Math III · A-CED.1

Creating and Solving One-Variable Equations and Inequalities Across Advanced Expression Types

This objective teaches students to turn real situations into solvable equations even when the relationship is polynomial, rational, radical, exponential, or otherwise more advanced than linear.

Concept Algebra

Domain Creating Equations

Read time 8 minutes

What this learning objective is really asking you to learn

This objective asks students to create and solve one-variable equations and inequalities from contexts using the expression types studied across high school. In Math I, students created linear equations and inequalities from real situations. In Math II, they added quadratics, simple rational expressions, exponential models, radical notation, and complex solutions. In Math III, the modeling toolkit expands again. Students may now need to create equations involving polynomials, rational expressions, radical or root functions, exponentials, and other familiar structures.

The heart of the objective is still modeling. A context describes a relationship. The student must identify the unknown, define a variable, choose the mathematical structure that fits the situation, write an equation or inequality, solve it, and interpret the answer. The expressions are more advanced, but the modeling cycle is the same.

For example, if a square has area 80 square units and side length $s$, the equation is

\[s^2 = 80\].

Solving gives $s = \sqrt{80} = 4\sqrt{5}$, and the negative solution is rejected because length cannot be negative.

If an average cost model is $(500 + 12x)/x = 20$, students solve a rational equation. If a falling object has height $h(t) = -16t^2 + 64t + 80$, students solve a quadratic equation or inequality to determine when the height reaches a target or stays above the ground. If a root function describes a physical relationship, students may need to solve an equation involving a square root and check for extraneous solutions.

This objective is asking students to become flexible modelers. They should not ask, “Which chapter is this?” They should ask, “What relationship does the situation describe, and what kind of equation represents it?”

Why students should learn this math

Students should learn this because real problems do not announce their algebra type. A business problem may involve a rational expression. A geometry problem may involve a square root. A physics problem may involve a quadratic. A growth problem may involve an exponential. A design problem may involve a polynomial. Adult quantitative reasoning requires choosing the model, not just solving a pre-labeled equation.

This is a major difference between early algebra and mature algebra. Early algebra often gives students equations to solve. Modeling asks students to create the equation. That is harder and more valuable. Writing the equation correctly is often the real work.

For example, suppose a company has a fixed setup cost of $1,200 and variable cost of $8 per unit. If the company wants average cost to be at most $12 per unit, the model is

\[(1200 + 8x)/x \le 12\].

Solving this tells how many units must be produced to bring average cost down. This is rational-expression modeling, and it represents a real economic idea: fixed cost spread across more units.

Suppose a sensor's response time is modeled by $t = \sqrt{d/5}$, and the maximum allowed response time is 3 seconds. Then

\[\sqrt{d/5} \le 3\].

Solving gives $d/5 \le 9$, so $d \le 45$. This is radical or root-function modeling.

The “why” is that advanced expression types represent different real mechanisms. Quadratics represent area, acceleration, and turning behavior. Rational expressions represent ratios and averages. Radical functions represent inverse power relationships. Exponentials represent repeated multiplicative change. Students need to match the mechanism to the model.

The historical machinery: algebra as a modeling language

Algebra developed from practical problems involving unknown quantities. Over time, the class of equations expanded. Linear equations handled constant-rate and balance problems. Quadratics handled area and projectile-like relationships. Polynomials handled more complex curves. Rational expressions handled ratios. Radical expressions handled inverse power relationships. Exponentials handled growth and decay.

The history of algebra is not only the history of solving techniques. It is also the history of expanding what kinds of situations can be represented. Each new expression type adds modeling power. A student who can only write linear equations has a limited modeling language. A student who can choose among linear, quadratic, polynomial, rational, radical, and exponential forms has a much broader toolkit.

Modern science and technology rely on this flexibility. Models are selected based on mechanism, data, and assumptions. A formula is not chosen because it is convenient; it is chosen because it captures the relationship. This objective is a school-level version of that modeling practice.

Where this fits in the big map of mathematics

This objective begins the Math III Creating Equations group. It revisits the very first high-school modeling skill but at a more advanced level. Objective 001 asked students to create and solve one-variable equations and inequalities from real situations. Objective 140 asks them to do that using all expression types they have now studied.

It connects backward to rational expressions, radicals, quadratics, exponentials, and polynomial functions. It connects forward to creating equations in two or more variables, systems of constraints, formula rearrangement, and approximate solving with advanced functions.

It connects to modeling standards across the course. The student must define quantities, choose units, interpret viable solutions, and reject extraneous answers.

It connects to domain restrictions. Rational equations may exclude denominator-zero values. Radical equations may require nonnegative radicands. Context may reject negative lengths, fractional people, or times outside the modeled interval.

The big-map role is integration. Students are no longer learning isolated equation types; they are choosing from a full algebraic toolbox.

How to execute the skill technically

Use a modeling routine:

Read the context and identify the unknown.
Define the variable with units.
Identify the relationship type.
Write the equation or inequality.
Solve using appropriate algebra.
Check for extraneous or nonviable solutions.
Interpret the answer in context.

Example: A rectangular garden has length 5 meters more than its width. Its area is 84 square meters. Find the dimensions.

Let $w$ be the width. Then length is $w + 5$. Area is

\[w(w + 5) = 84\].

\[w^2 + 5w - 84 = 0\].

Factor:

\[(w + 12)(w - 7) = 0\].

So $w = -12$ or $w = 7$. Reject -12 because width cannot be negative. The width is 7 meters, and the length is 12 meters.

Example: A company's average cost is modeled by

\[A(x) = (900 + 15x)/x\].

How many units must be produced for average cost to be at most $25?

Solve

\[(900 + 15x)/x \le 25\].

Assuming $x > 0$, multiply by $x$ without reversing the inequality:

\[900 + 15x \le 25x\].

900 \le 10x

and

\[x \ge 90\].

The company must produce at least 90 units.

This example shows why assumptions matter. Since $x$ is number of units, $x > 0$, so multiplying by $x$ is safe.

Worked example: radical equation from geometry

The diagonal $d$ of a square with side length $s$ is

\[d = s\sqrt{2}\].

Suppose a square screen has diagonal 20 inches. Find the side length.

Let $s$ be the side length in inches. Then

\[s\sqrt{2} = 20\].

Divide by $\sqrt{2}$:

\[s = 20/\sqrt{2} = 10\sqrt{2}\].

So the side length is $10\sqrt{2}$ inches, about 14.14 inches.

Now consider a square with area $A$. Its side length is $s = \sqrt{A}$. If a design requires side length at least 12, then

\[\sqrt{A} \ge 12\].

Since both sides are nonnegative, square both sides:

\[A \ge 144\].

The area must be at least 144 square units.

Extraneous solutions and context checks

Advanced equation types make checking essential. Squaring both sides of a radical equation can introduce extraneous solutions. Multiplying by an expression in a rational equation can include values that make the original denominator zero. Solving a polynomial equation can produce negative values that fail a physical context.

For example, solving $\sqrt{x + 5} = x - 1$ requires $x - 1 \ge 0$, so $x \ge 1$. Squaring gives $x + 5 = x^2 - 2x + 1$, or $x^2 - 3x - 4 = 0$, so $x = 4$ or $x = -1$. But $x = -1$ fails the original equation because the right side is -2 while the square root is nonnegative. Only $x = 4$ works.

This is why Math III equation solving must end with checking.

Choosing the expression type from the situation

A mature algebra student learns to identify the mechanism.

Use a linear model when there is constant additive change: a fixed fee plus a constant rate.

Use a quadratic model when the situation involves area from two related lengths, projectile motion, or a single turning point.

Use a polynomial model when several dimensions or effects multiply, or when a curve has more complex turning behavior.

Use a rational expression when the situation involves a ratio, average, rate, or variable in the denominator.

Use a radical or root function when a relationship undoes a power, such as finding side length from area or radius from volume.

Use an exponential model when change happens by a constant factor or percent over equal intervals.

This classification is one of the most important parts of the objective. The equation does not come with a label in the real world. The student must infer the model from the relationship.

Worked example: exponential context

A bacteria culture starts with 500 bacteria and doubles every hour. When will it reach at least 8,000 bacteria?

Let $t$ be hours. The model is

\[500 \cdot 2^t \ge 8000\].

Divide by 500:

\[2^t \ge 16\].

Since $16 = 2^4$, we get

\[t \ge 4\].

The culture reaches at least 8,000 bacteria after 4 hours.

This is a simple exponential inequality. It is different from a linear model because the culture doubles; it does not add the same number each hour.

Worked example: rational context with extraneous restrictions

A pipe fills a tank in $x$ hours. A second pipe takes 2 hours longer, so its time is $x + 2$. Together, their combined rate is

\[1/x + 1/(x + 2)\].

If together they fill the tank in 3 hours, their combined rate is $1/3$, so

\[1/x + 1/(x + 2) = 1/3\].

The domain requires $x > 0$. Multiply by $3x(x + 2)$:

\[3(x + 2) + 3x = x(x + 2)\].

\[6x + 6 = x^2 + 2x\].

Rearrange:

\[x^2 - 4x - 6 = 0\].

Solving gives

\[x = 2 ± \sqrt{10}\].

Only $x = 2 + \sqrt{10}$ is positive, so that is the valid time. The negative solution is rejected. This example shows how context and domain control the final answer.

Modeling inequalities

Equations describe exact targets. Inequalities describe acceptable ranges. In real life, inequalities are often more realistic. A bridge must support at least a load. A cost must be no more than a budget. A medicine concentration must stay within a safe range. A production level must meet or exceed demand.

For example, if average cost is $(1000 + 20x)/x$ and the target is no more than 30 dollars per unit, the model is

\[(1000 + 20x)/x \le 30\].

Assuming $x > 0$, solving gives

1000 + 20x \le 30x

\[x \ge 100\].

The interpretation is that at least 100 units must be produced to meet the average-cost target. The inequality solution is a range, not one number.

Common misconceptions and how to avoid them

One common mistake is trying to force every context into a linear model. Different mechanisms require different expression types.

Another mistake is solving correctly but failing to interpret the answer. Context determines whether a solution is viable.

A third mistake is forgetting restrictions in rational equations. Denominators cannot be zero.

A fourth mistake is accepting extraneous solutions from radical equations without checking.

A fifth mistake is multiplying or dividing inequalities by variable expressions without knowing whether they are positive or negative.

The big takeaway

This objective asks students to use the full algebraic toolbox to model one-variable situations. The skill is not merely solving; it is choosing the right expression type, writing the equation or inequality, solving carefully, checking restrictions, and interpreting the result. This is mature algebraic modeling.

Problem Library

Problems in the App From This Objective

186 problems across 15 archetypes in the app.

model with higher-degree polynomial and find valid roots.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Create and solve a polynomial equation from context: box volume x(x+2)(x+3)=60.

Problem 2

Create and solve a polynomial equation from context: revenue x(100-2x)=1200.

Problem 3

Create and solve a polynomial equation from context: projectile height -16t^2+64t=0.

Problem 4

Create and solve a polynomial equation from context: higher-degree context.

Problem 5

Create and solve a polynomial equation from context: A rectangular garden has a length that is 5 feet more than its width. If the area of the garden is 84 square feet.

Problem 6

Create and solve a polynomial equation from context: The product of three consecutive integers is 210.

Problem 7

Create and solve a polynomial equation from context: The legs of a right triangle are x and x+7. The hypotenuse is x+8.

Problem 8

Create and solve a polynomial equation from context: An object is dropped from a height of 100 feet. Its height h (in feet) after t seconds is given by h = 100 - 16t^2. Find the time it takes for the object to hit the ground.

Problem 9

Create and solve a polynomial equation from context: A cylindrical can has a height that is 3 times its radius. If its volume is 243π cubic units.

Problem 10

Create and solve a polynomial equation from context: A company's profit P (in thousands of dollars) from selling x thousand items is given by P = -x^3 + 12x^2 - 20x. Find the number of items sold to achieve a profit of 64 thousand dollars.

Open in simulator

Problem 11

Create and solve a polynomial equation from context: A rectangular garden is 10 feet by 15 feet. A path of uniform width x is built around the garden. If the total area of the garden and path is 250 square feet.

Problem 12

Create and solve a polynomial equation from context: The difference between the cube of a number and twice its square is 48.

model rates or ratios and check restrictions.

15 problems Warmup Practice Mixed Review Assessment

Problem 13

Create and solve a rational equation from context: two workers take x and x+2 hours, combined rate completes 1 job in 3 hours.

Problem 14

Create and solve a rational equation from context: average cost C(x)=500/x+12 reaches 20.

Problem 15

Create and solve a rational equation from context: travel time d/r plus d/(r+10) equals total time.

Problem 16

Create and solve a rational equation from context: rate or ratio context.

Open in simulator

Problem 17

Create and solve a rational equation from context: A pipe fills a tank in x hours, another drains it in x+5 hours. If both are open, the tank fills in 6 hours.

Problem 18

Create and solve a rational equation from context: A boat travels 30 miles upstream and 30 miles downstream. The current is 5 mph. Total trip takes 8 hours. Find boat speed in still water.

Problem 19

Create and solve a rational equation from context: A car travels 100 miles at speed x, then 150 miles at speed x+10. The total time taken is 5 hours.

Problem 20

Create and solve a rational equation from context: Two similar triangles have corresponding sides in ratio x:(x+3). The ratio of their areas is 4:9.

Problem 21

Create and solve a rational equation from context: The cost to produce x items is C(x) = 1000 + 5x. The average cost per item is $15.

Problem 22

Create and solve a rational equation from context: A 20-liter solution is 30% acid. How much pure acid (100%) must be added to make it 50% acid?.

Problem 23

Create and solve a rational equation from context: Worker A takes x hours, B takes x-2 hours, C takes x+1 hours. Together they complete a job in 2 hours.

Problem 24

Create and solve a rational equation from context: A person drives 120 miles. For the first half of the distance, they drive at speed x. For the second half, they drive at speed x-10 mph. The total trip takes 3 hours.

Problem 25

Create and solve a rational equation from context: A recipe calls for 3 cups of flour for every 2 cups of sugar. If you have x cups of flour and x-1 cups of sugar, and you want to maintain the ratio.

Problem 26

Create and solve a rational equation from context: Two resistors R1 and R2 are connected in parallel. R1 is x ohms, R2 is x+2 ohms. The total equivalent resistance is 1.5 ohms.

Problem 27

Create and solve a rational equation from context: A cyclist rides 40 miles to a destination and returns. The speed on the return trip is 5 mph slower than the speed to the destination. The total time for the round trip is 6 hours.

model square-root relationship and check extraneous solutions.

12 problems Warmup Practice Mixed Review Assessment

Problem 28

Create and solve a radical equation from context: distance from origin sqrt(x^2+9)=5.

Problem 29

Create and solve a radical equation from context: side length sqrt(A)=12.

Problem 30

Create and solve a radical equation from context: speed model v=sqrt(2gh) reaches target v.

Problem 31

Create and solve a radical equation from context: geometry square-root relationship.

Problem 32

Create and solve a radical equation from context: A right triangle has a hypotenuse of 10 cm and one leg of 6 cm. Determine the length of the other leg.

Problem 33

Create and solve a radical equation from context: A simple pendulum completes one swing (period) in 3 seconds. Given g = 9.8 m/s^2, calculate the length of the pendulum using the formula T = 2π√(L/g).

Problem 34

Create and solve a radical equation from context: The distance between point A(2, 3) and point B(x, 7) is 5 units. Find the possible x-coordinates for point B.

Problem 35

Create and solve a radical equation from context: A problem states that the square root of a number, when multiplied by 2, is 10. Find the number.

Open in simulator

Problem 36

Create and solve a radical equation from context: The velocity (v) of an object dropped from a height (h) is given by v = √(2gh). If an object hits the ground at 14 m/s and g = 9.8 m/s^2, from what height was it dropped?.

Problem 37

Create and solve a radical equation from context: The geometric mean of two positive numbers a and b is √(ab). If the geometric mean of 4 and another number x is 6, find x.

Problem 38

Create and solve a radical equation from context: In an AC circuit, the impedance (Z) is given by Z = √(R^2 + X^2), where R is resistance and X is reactance. If the impedance is 25 ohms and the resistance is 7 ohms, find the reactance X.

Problem 39

Create and solve a radical equation from context: Consider a scenario where the square root of the sum of a number and 5 is equal to the number itself minus 1. Find the number.

model growth/decay and solve with logs or technology.

15 problems Warmup Practice Mixed Review Assessment

Problem 40

Create and solve an exponential equation from context: population 500 grows by 8% to 1000.

Problem 41

Create and solve an exponential equation from context: half-life model 80(1/2)^(t/6)=10.

Open in simulator

Problem 42

Create and solve an exponential equation from context: investment 1000(1.05)^t=1500.

Problem 43

Create and solve an exponential equation from context: growth or decay context.

Problem 44

Create and solve an exponential equation from context: bacteria population of 200 doubles every 3 hours, reaches 3200.

Problem 45

Create and solve an exponential equation from context: car worth $20,000 depreciates by 15% annually to $10,000.

Problem 46

Create and solve an exponential equation from context: initial investment of $5000 with 6% annual interest compounded monthly reaches $7500.

Problem 47

Create and solve an exponential equation from context: a substance with half-life of 5 years decays from 64 grams to 2 grams.

Problem 48

Create and solve an exponential equation from context: an initial amount of 100 grows by 12% per period to 500.

Problem 49

Create and solve an exponential equation from context: a substance at 100 degrees cools such that its temperature decreases by 5% every minute, reaching 70 degrees.

Problem 50

Create and solve an exponential equation from context: a population triples every 4 years, starting at 100 and reaching 2700.

Problem 51

Create and solve an exponential equation from context: solve for t in the equation 250 * (1.03)^t = 400.

Problem 52

Create and solve an exponential equation from context: solve for x in the equation 120 * (0.9)^x = 50.

Problem 53

Create and solve an exponential equation from context: a radioactive isotope has a half-life of 10 days. If you start with 128 grams, how long until only 4 grams remain?.

Problem 54

Create and solve an exponential equation from context: an investment of $2000 grows at 7% annually to $4000.

translate inverse exponential relationship.

12 problems Warmup Practice Mixed Review Assessment

Problem 55

Create and solve a logarithmic equation from context: pH=-log(H) and pH is 3.

Problem 56

Create and solve a logarithmic equation from context: decibels D=10log(I/I0), given D target.

Problem 57

Create and solve a logarithmic equation from context: time-to-target from log model t=log(A/B)/k.

Problem 58

Create and solve a logarithmic equation from context: inverse exponential relationship.

Problem 59

Create and solve a logarithmic equation from context: Richter scale R=log(I/I0) where I is intensity and I0 is reference intensity. An earthquake has R=6.5.

Problem 60

Create and solve a logarithmic equation from context: pH=-log[H+] and a solution has pH of 7.4.

Problem 61

Create and solve a logarithmic equation from context: Population P grows according to P=P0*e^(kt). Find time t for P to double if k=0.05.

Problem 62

Create and solve a logarithmic equation from context: Continuously compounded interest A=Pe^(rt). Find time t for an investment to reach 3 times its initial value at r=0.06.

Problem 63

Create and solve a logarithmic equation from context: Decibel level D=10log(I/I0). A sound has D=80.

Open in simulator

Problem 64

Create and solve a logarithmic equation from context: Radioactive decay N=N0*e^(-λt). Find half-life t if λ=0.001 per year.

Problem 65

Create and solve a logarithmic equation from context: Apparent magnitude m = C - 2.5log(L), where L is luminosity. Find L if m=5 and C=10.

Problem 66

Create and solve a logarithmic equation from context: Solve for x in the equation log_b(x) = y, given b=5 and y=3.

model distance/tolerance and split cases.

12 problems Warmup Practice Mixed Review Assessment

Problem 67

Create and solve an absolute-value equation or inequality from context: measurement must be within 0.5 of 10.

Problem 68

Create and solve an absolute-value equation or inequality from context: temperature differs from 72 by exactly 4.

Problem 69

Create and solve an absolute-value equation or inequality from context: error exceeds 3 units from target 100.

Problem 70

Create and solve an absolute-value equation or inequality from context: distance or tolerance context.

Open in simulator

Problem 71

Create and solve an absolute-value equation or inequality from context: A machine's output must be less than 0.1 units from its ideal setting of 50.

Problem 72

Create and solve an absolute-value equation or inequality from context: The concentration of a chemical must be at least 2 units away from the optimal level of 15.

Problem 73

Create and solve an absolute-value equation or inequality from context: The weight of a product must be exactly 0.2 kg from its standard weight of 5 kg.

Problem 74

Create and solve an absolute-value equation or inequality from context: The height of a component can deviate from 12 cm by no more than 0.05 cm.

Problem 75

Create and solve an absolute-value equation or inequality from context: The speed of a vehicle must be more than 10 mph away from the average speed of 60 mph.

Problem 76

Create and solve an absolute-value equation or inequality from context: The actual volume differs from the target volume of 250 ml by no less than 5 ml.

Problem 77

Create and solve an absolute-value equation or inequality from context: The age of a historical artifact is known to be 2000 years, but its actual age differs from this by more than 50 years.

Problem 78

Create and solve an absolute-value equation or inequality from context: The acceptable range for a pH level is between 6.8 and 7.2.

apply domain restriction and solve boundary.

12 problems Warmup Practice Mixed Review Assessment

Problem 79

Create and solve a root-function inequality from context: sqrt(x-3) <= 5.

Open in simulator

Problem 80

Create and solve a root-function inequality from context: sqrt(h) must be at least 4.

Problem 81

Create and solve a root-function inequality from context: cube-root model cbrt(x-2)>3.

Problem 82

Create and solve a root-function inequality from context: square-root model with boundary target.

Problem 83

Create and solve a root-function inequality from context: sqrt(x+1) < -2.

Problem 84

Create and solve a root-function inequality from context: sqrt(2x-4) > -1.

Problem 85

Create and solve a root-function inequality from context: sqrt(x/2 + 1) >= 3.

Problem 86

Create and solve a root-function inequality from context: sqrt(x^2 - 9) <= 4.

Problem 87

Create and solve a root-function inequality from context: cbrt(5x + 7) <= 2.

Problem 88

Create and solve a root-function inequality from context: root(4, x-1) > 2.

Problem 89

Create and solve a root-function inequality from context: 2*sqrt(x+5) < 10.

Problem 90

Create and solve a root-function inequality from context: 3 < sqrt(x-2) <= 5.

distinguish polynomial, rational, radical, exponential, logarithmic, and absolute-value structures.

12 problems Warmup Practice Mixed Review Assessment

Problem 91

Choose the appropriate one-variable model type for context quantity grows by the same percent each year.

Problem 92

Choose the appropriate one-variable model type for context total cost divided by number of items.

Problem 93

Choose the appropriate one-variable model type for context distance from target must be within tolerance.

Problem 94

Choose the appropriate one-variable model type for context area or volume built from polynomial dimensions.

Problem 95

Choose the appropriate one-variable model type for context cost increases by a fixed amount for each additional unit.

Problem 96

Choose the appropriate one-variable model type for context height of an object thrown upwards over time.

Problem 97

Choose the appropriate one-variable model type for context period of a pendulum based on its length.

Open in simulator

Problem 98

Choose the appropriate one-variable model type for context combined rate of work for multiple individuals.

Problem 99

Choose the appropriate one-variable model type for context time to reach a specific population size given continuous growth.

Problem 100

Choose the appropriate one-variable model type for context radioactive decay of a substance over time.

Problem 101

Choose the appropriate one-variable model type for context cost structure changes at a certain quantity threshold.

Problem 102

Choose the appropriate one-variable model type for context temperature variation from a set point must be within a range.

reject extraneous or nonviable roots.

12 problems Warmup Practice Mixed Review Assessment

Problem 103

Interpret all algebraic solutions in context for time solutions t=-2 and t=5.

Problem 104

Interpret all algebraic solutions in context for length solutions x=0 and x=8 for positive side length.

Problem 105

Interpret all algebraic solutions in context for radical equation gives x=1 and x=9, but x=1 fails original equation.

Problem 106

Interpret all algebraic solutions in context for money amount solutions -50 and 200.

Problem 107

Interpret all algebraic solutions in context for time solutions t=-3 and t=7.

Problem 108

Interpret all algebraic solutions in context for distance solutions d=-10 and d=50.

Problem 109

Interpret all algebraic solutions in context for population solutions P=-100 and P=1500.

Problem 110

Interpret all algebraic solutions in context for age solutions a=-5 and a=25.

Problem 111

Interpret all algebraic solutions in context for logarithmic equation gives x=0 and x=4, but x=0 is outside log domain.

Open in simulator

Problem 112

Interpret all algebraic solutions in context for rational equation gives x=2 and x=5, but x=2 causes a zero denominator.

Problem 113

Interpret all algebraic solutions in context for speed solutions -15 mph and 20 mph.

Problem 114

Interpret all algebraic solutions in context for area solutions A=-10 and A=100.

model physical limits and solve equation/inequality.

12 problems Warmup Practice Mixed Review Assessment

Problem 115

Create and solve a one-variable design constraint for open box volume x(20-2x)(30-2x)=1000.

Problem 116

Create and solve a one-variable design constraint for fencing cost 12x+8(100-x)<=1000.

Problem 117

Create and solve a one-variable design constraint for cylindrical can volume pi r^2 h=500 with h=2r.

Problem 118

Create and solve a one-variable design constraint for material area must not exceed budget B.

Problem 119

Create and solve a one-variable design constraint for rectangular garden area at least 400 sq ft with 100 ft perimeter.

Problem 120

Create and solve a one-variable design constraint for rectangular enclosure against a wall, 3 sides fenced, total fence length 200m, maximize area.

Problem 121

Create and solve a one-variable design constraint for conical container volume at least 100 cubic cm with slant height 10cm.

Open in simulator

Problem 122

Create and solve a one-variable design constraint for manufacturing cost per unit (0.01x^2 - 2x + 500)/x must be <= $10 for x units.

Problem 123

Create and solve a one-variable design constraint for mixing 10L of 20% acid with x L of 50% acid to get at least 30% acid.

Problem 124

Create and solve a one-variable design constraint for ball thrown upwards h(t)=30t-4.9t^2, must reach at least 40m.

Problem 125

Create and solve a one-variable design constraint for circular sector with perimeter 20cm, maximize area.

Problem 126

Create and solve a one-variable design constraint for budget $1000 for x units of item A ($50) and y units of item B ($30), where y=2x.

infer model and solve target condition.

12 problems Warmup Practice Mixed Review Assessment

Problem 127

Create an equation from graph or table data and solve for target: table doubles each step from 3 to 6 to 12; find when value is 48.

Problem 128

Create an equation from graph or table data and solve for target: graph shows polynomial crossing target y=0 at x=2 and x=5.

Problem 129

Create an equation from graph or table data and solve for target: reciprocal table has xy=12; solve y=3.

Problem 130

Create an equation from graph or table data and solve for target: target horizontal line intersects model graph.

Problem 131

Create an equation from graph or table data and solve for target: table shows (0, 5), (1, 7), (2, 9); find x when y is 15.

Problem 132

Create an equation from graph or table data and solve for target: graph of a parabola opens up with vertex at (1, -2) and passes through (0, -1); find x when y is 7.

Problem 133

Create an equation from graph or table data and solve for target: table shows (0, 100), (1, 50), (2, 25); find when value is 12.5.

Problem 134

Create an equation from graph or table data and solve for target: graph of an absolute value function has vertex at (3, 0) and passes through (0, 3); find x when y is 5.

Problem 135

Create an equation from graph or table data and solve for target: table shows (1, 0), (e, 1), (e^2, 2); find x when y is 3.

Problem 136

Create an equation from graph or table data and solve for target: graph has vertical asymptote at x=2, horizontal asymptote at y=1, and passes through (3, 4); find x when y is 2.

Open in simulator

Problem 137

Create an equation from graph or table data and solve for target: graph shows a sine wave with amplitude 1, period 2pi, passing through (0,0) and (pi/2, 1); find x when y is 0.5 for x in [0, pi/2].

Problem 138

Create an equation from graph or table data and solve for target: graph shows a line passing through (0, -3) and (2, 1); find x when y is 5.

identify better structure and constraints.

12 problems Warmup Practice Mixed Review Assessment

Problem 139

Compare candidate one-variable models for context population grows by 6% per year.

Problem 140

Compare candidate one-variable models for context average cost equals fixed cost divided by items plus unit cost.

Problem 141

Compare candidate one-variable models for context distance from target tolerance.

Problem 142

Compare candidate one-variable models for context area from side expressions.

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Problem 143

Compare candidate one-variable models for context height of a projectile launched upwards over time.

Problem 144

Compare candidate one-variable models for context amount of a radioactive isotope remaining after a certain number of half-lives.

Problem 145

Compare candidate one-variable models for context total earnings from an hourly wage.

Problem 146

Compare candidate one-variable models for context resistance of a wire as its cross-sectional area changes.

Problem 147

Compare candidate one-variable models for context daily temperature fluctuations throughout a year.

Problem 148

Compare candidate one-variable models for context concentration of a drug in the body after an initial rapid absorption and subsequent slower elimination.

Problem 149

Compare candidate one-variable models for context volume of a sphere given its radius.

Problem 150

Compare candidate one-variable models for context value of an investment with compound interest over time.

set up equation and interpret numerical solution.

12 problems Warmup Practice Mixed Review Assessment

Problem 151

Use technology-supported solving for advanced model x^5-x-10=0 in context x>0.

Problem 152

Use technology-supported solving for advanced model 1000(1.03)^t=2000.

Problem 153

Use technology-supported solving for advanced model sqrt(x+4)+x=10.

Problem 154

Use technology-supported solving for advanced model nonfactorable equation f(x)=target.

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Problem 155

Use technology-supported solving for advanced model x^4 - 3x^2 + x - 5 = 0 for x < 0.

Problem 156

Use technology-supported solving for advanced model 50 * e^(0.05t) = 500.

Problem 157

Use technology-supported solving for advanced model ln(x) + x = 5.

Problem 158

Use technology-supported solving for advanced model sin(x) = x/2 for x > 0.

Problem 159

Use technology-supported solving for advanced model e^x = x^2 + 2.

Problem 160

Use technology-supported solving for advanced model cbrt(x^2+5) + x = 7.

Problem 161

Use technology-supported solving for advanced model A population grows according to P(t) = 1000 / (1 + 9e^(-0.1t)). When does P(t) = 800?.

Problem 162

Use technology-supported solving for advanced model 2x^3 - 4x^2 + x - 5 = 0 for x in [2, 3].

identify denominators, radicands, logs, and context limits.

12 problems Warmup Practice Mixed Review Assessment

Problem 163

State domain restrictions before solving modeled equation 1/(x-3)=5.

Problem 164

State domain restrictions before solving modeled equation sqrt(x-2)=7.

Problem 165

State domain restrictions before solving modeled equation log(x+4)=2.

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Problem 166

State domain restrictions before solving modeled equation context length x in box design.

Problem 167

State domain restrictions before solving modeled equation 1/(2x+1)=3.

Problem 168

State domain restrictions before solving modeled equation 5/(x^2-9)=1.

Problem 169

State domain restrictions before solving modeled equation sqrt(5-x)=10.

Problem 170

State domain restrictions before solving modeled equation sqrt(2x-6)=4.

Problem 171

State domain restrictions before solving modeled equation ln(2x-1)=3.

Problem 172

State domain restrictions before solving modeled equation log_2(x^2)=4.

Problem 173

State domain restrictions before solving modeled equation context radius r of a sphere.

Problem 174

State domain restrictions before solving modeled equation context number of students s in a class.

catch model type, domain, extraneous solution, and interpretation mistakes.

12 problems Warmup Practice Mixed Review Assessment

Problem 175

Correct the advanced modeling equation error: A student uses a linear model for constant percent growth.

Problem 176

Correct the advanced modeling equation error: A student keeps a negative time solution.

Problem 177

Correct the advanced modeling equation error: A student squares a radical equation and keeps an extraneous root.

Problem 178

Correct the advanced modeling equation error: A student solves a rational model but allows zero denominator.

Problem 179

Correct the advanced modeling equation error: A student uses a logarithmic model and accepts a solution that results in the logarithm of a non-positive number.

Problem 180

Correct the advanced modeling equation error: A student models a periodic phenomenon with a sine function but calculates a value outside the function's range, like a negative height.

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Problem 181

Correct the advanced modeling equation error: A student incorrectly interprets the vertex of a quadratic profit function as the minimum profit when it represents the maximum.

Problem 182

Correct the advanced modeling equation error: A student solves an absolute value equation by squaring both sides and accepts an extraneous solution.

Problem 183

Correct the advanced modeling equation error: A student finds the inverse of a function but doesn't restrict its domain to match the original function's range in context.

Problem 184

Correct the advanced modeling equation error: A student attempts to find a unique intersection point for two parallel lines in a system of equations, expecting a single solution.

Problem 185

Correct the advanced modeling equation error: A student uses the wrong sub-function of a piecewise model for a given input value.

Problem 186

Correct the advanced modeling equation error: A student models growth in centimeters per year but interprets the result as meters per year without conversion.