chapter 2 equations inequalities and problem solving answers

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 2: Solving equations & inequalities

About this unit, linear equations with variables on both sides.

• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
• Intro to equations with variables on both sides (Opens a modal)
• Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
• Equation with variables on both sides: fractions (Opens a modal)
• Equation with the variable in the denominator (Opens a modal)
• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

Linear equations with parentheses

• Equations with parentheses (Opens a modal)
• Reasoning with linear equations (Opens a modal)
• Multi-step equations review (Opens a modal)
• Equations with parentheses Get 3 of 4 questions to level up!
• Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
• Reasoning with linear equations Get 3 of 4 questions to level up!

Analyzing the number of solutions to linear equations

• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Linear equations with unknown coefficients

• Linear equations with unknown coefficients (Opens a modal)
• Why is algebra important to learn? (Opens a modal)
• Linear equations with unknown coefficients Get 3 of 4 questions to level up!

Multi-step inequalities

• Inequalities with variables on both sides (Opens a modal)
• Inequalities with variables on both sides (with parentheses) (Opens a modal)
• Multi-step inequalities (Opens a modal)
• Using inequalities to solve problems (Opens a modal)
• Multi-step linear inequalities Get 3 of 4 questions to level up!
• Using inequalities to solve problems Get 3 of 4 questions to level up!

Compound inequalities

• Compound inequalities: OR (Opens a modal)
• Compound inequalities: AND (Opens a modal)
• A compound inequality with no solution (Opens a modal)
• Double inequalities (Opens a modal)
• Compound inequalities examples (Opens a modal)
• Compound inequalities review (Opens a modal)
• Solving equations & inequalities: FAQ (Opens a modal)
• Compound inequalities Get 3 of 4 questions to level up!

Introduction to Equations and Inequalities

Chapter outline.

Irrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and productive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related equipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation systems and conduits can be installed within the plotted area.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/2-introduction-to-equations-and-inequalities

© Jan 9, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Snapsolve any problem by taking a picture. Try it in the Numerade app?

Algebra and Trigonometry Real Mathematics, Real People

Solving equations and inequalities - all with video answers.

Linear Equations and Problem Solving

Fill in the blank.

A(n) _______ is a statement that two algebraic expressions are equal.

A linear equation in one variable is an equation that can be written in the standard form _______ .

When solving an equation, it is possible to introduce a(n) _______ solution, which is a value that does not satisfy the original equation.

Many real-life problems can be solved using ready-made equations called _______ .

Is the equation an identity, a conditional equation, or a contradiction?

How can you clear the equation $\frac{x}{2}+1=\frac{1}{4}$ of fractions?

Determine whether each value of $x$ is a solution of the equation. Equation $\frac{5}{2 x}-\frac{4}{x}=3$

Values (a) $x=-\frac{1}{2}$ (b) $x=4$ (c) $x=0$ (d) $x=\frac{1}{4}$

Determine whether each value of $x$ is a solution of the equation. Equation $\frac{x}{2}+\frac{6 x}{7}=\frac{19}{14}$

Values (a) $x=-2$ (b) $x=1$ (c) $x=\frac{1}{2}$ (d) $x=7$

Determine whether each value of $x$ is a solution of the equation. Equation $\frac{\sqrt{x+4}}{6}+3=4$ Values (a) $x=-3$ (b) $x=0$ (c) $x=21$ (d) $x=32$

Determine whether each value of $x$ is a solution of the equation. Equation $$\frac{\sqrt[3]{x-8}}{3}=-\frac{2}{3}$$ Values (a) $x=-16$ (b) $x=0$ (c) $x=9$ (d) $x=16$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$2(x-1)=2 x-2$$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$x^{2}-8 x+5=(x-4)^{2}-11$$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$-5(x-1)=-5(x+1)$$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$(x+3)(x-5)=x^{2}-2(x+7)$$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$3+\frac{1}{x+1}=\frac{4 x}{x+1}$$

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$\frac{5}{x}+\frac{3}{x}=24$$

Solve the equation using two methods. Then explain which method is easier. $$\frac{3 x}{8}-\frac{4 x}{3}=4$$

Solve the equation using two methods. Then explain which method is easier. $$\frac{3 z}{8}-\frac{z}{10}=6$$

Solve the equation using two methods. Then explain which method is easier. $$\frac{2 x}{5}+5 x=\frac{4}{3}$$

Solve the equation using two methods. Then explain which method is easier. $$\frac{4 y}{3}-2 y=\frac{16}{5}$$

Solve the equation (if possible). $$3 x-5=2 x+7$$

Solve the equation (if possible). $$5 x+3=6-2 x$$

Solve the equation (if possible). $$3(y-5)=3+5 y$$

Solve the equation (if possible). $$5(z-4)+4 z=5-6 z$$

Solve the equation (if possible). $$\frac{x}{5}-\frac{x}{2}=3$$

Solve the equation (if possible). $$\frac{5 x}{4}+\frac{1}{2}=x-\frac{1}{2}$$

Solve the equation (if possible). $$\frac{2(z-4)}{5}+5=10 z$$

Solve the equation (if possible). $$\frac{3 x}{2}+\frac{1}{4}(x-2)=10$$

Solve the equation (if possible). $$\frac{100-4 u}{3}=\frac{5 u+6}{4}+6$$

Solve the equation (if possible). $$\frac{17+y}{y}+\frac{32+y}{y}=100$$

Solve the equation (if possible). $$\frac{5 x-4}{5 x+4}=\frac{2}{3}$$

Solve the equation (if possible). $$\frac{10 x+3}{5 x+6}=\frac{1}{2}$$

Solve the equation (if possible). $$\frac{1}{x-3}+\frac{1}{x+3}=\frac{10}{x^{2}-9}$$

Solve the equation (if possible). $$\frac{1}{x-2}+\frac{3}{x+3}=\frac{4}{x^{2}+x-6}$$

Solve the equation (if possible). $$\frac{1}{x}+\frac{2}{x-5}=0$$

Solve the equation (if possible). $$3=2+\frac{2}{z+2}$$

Solve the equation (if possible). $$\frac{2}{(x-4)(x-2)}=\frac{1}{x-4}+\frac{2}{x-2}$$

Solve the equation (if possible). $$\frac{2}{x(x-2)}+\frac{5}{x}=\frac{1}{x-2}$$

Solve the equation (if possible). $$\frac{3}{x^{2}-3 x}+\frac{4}{x}=\frac{1}{x-3}$$

Solve the equation (if possible). $$\frac{6}{x}-\frac{2}{x+3}=\frac{3(x+5)}{x(x+3)}$$

Solve for the indicated variable. Area of a Triangle Solve for $h: A=\frac{1}{2} b h$

Solve for the indicated variable. Area of a Trapezoid Solve for $b: A=\frac{1}{2}(a+b) h$

Solve for the indicated variable. Investment at Compound Interest Solve for $P . A=P\left(1+\frac{r}{n}\right)^{n t}$

Solve for the indicated variable. Investment at Simple Interest Solve for $r . A=P+P r t$

Solve for the indicated variable. Volume of a Right Circular Cylinder Solve for $h: V=\pi r^{2} h$

Solve for the indicated variable. Volume of a Right Circular Cone Solve for $h: V=\frac{1}{3} \pi r^{2} h$

Use the following information. The relationship between the length of an adult's femur (thigh bone) and the height of the adult can be approximated by the linear equations $y=0.432 x-10.44$ Female $y=0.449 x-12.15 \quad$ Male where $y$ is the length of the femur in inches and $x$ is the height of the adult in inches (see figure). An anthropologist discovers a femur belonging to an adult human female. The bone is 16 inches long. Estimate the height of the female.

Use the following information. The relationship between the length of an adult's femur (thigh bone) and the height of the adult can be approximated by the linear equations $y=0.432 x-10.44$ Female $y=0.449 x-12.15 \quad$ Male where $y$ is the length of the femur in inches and $x$ is the height of the adult in inches (see figure). From the foot bones of an adult human male, an anthropologist estimates that the person's height was 69 inches. A few feet away from the site where the foot bones were discovered, the anthropologist discovers a male adult femur that is 19 inches long. Is it likely that both the foot bones and the thigh bone came from the same person?

A room is 1.5 times as long as it is wide, and its perimeter is 25 meters. (a) Draw a diagram that gives a visual representation of the problem. Identify the length as $I$ and the width as $w$ (b) Write $I$ in terms of $w$ and write an equation for the perimeter in terms of $w$ (c) Find the dimensions of the room.

A picture frame has a total perimeter of 3 meters. The height of the frame is $\frac{2}{3}$ times its width. (a) Draw a diagram that gives a visual representation of the problem. Identify the width as $w$ and the height as $h$ (b) Write $h$ in terms of $w$ and write an equation for the perimeter in terms of $w$ (c) Find the dimensions of the picture frame.

To get an A in a course, you must have an average of at least 90 on four tests of 100 points each. The scores on your first three tests were $87,92,$ and 84 (a) Write a verbal model for the test average for the course. (b) What is the least you can score on the fourth test to get an $\mathrm{A}$ in the course?

A store generates Monday through Thursday sales of $\$ 150, \$ 125, \$ 75,$ and $\$ 180 .$ What sales on Friday would give a weekday average of $\$ 150 ?$

A salesperson is driving from the office to a client, a distance of about 250 kilometers. After 30 minutes, the salesperson passes a town that is 50 kilometers from the office. Assuming the salesperson continues at the same constant speed, how long will it take to drive from the office to the client?

On the first part of a 336 -mile trip, a salesperson averaged 58 miles per hour. The salesperson averaged only 52 miles per hour on the last part of the trip because of an increased volume of traffic. The total time of the trip was 6 hours. Find the amount of time at each of the two speeds.

A truck driver traveled at an average speed of 55 miles per hour on a 200 -mile trip to pick up a load of freight. On the return trip (with the truck fully loaded), the average speed was 40 miles per hour. Find the average speed for the round trip.

An executive flew in the meeting in a city 1500 kilometers a the same amount of time on the return flight, the pilot mentioned that they still had 300 kilometers to go. The air speed of the plane was 600 kilometers per hour. How fast was the wind blowing? (Assume that the wind direction was parallel to the flight path and constant all day.)

To determine the height of a pine tree, you measure the shadow cast by the tree and find it to be 20 feet long (see figure). Then you measure the shadow cast by a 36 -inch tall oak sapling and find it to be 24 inches long. Estimate the pine tree's height.

A person who is 6 feet tall walks away from a flagpole toward the tip of the shadow of the flagpole. When the person is 30 feet from the flagpole, the tips of the person's shadow and the shadow cast by the flagpole coincide at a point 5 feet in front of the person. (a) Draw a diagram that illustrates the problem. Let $h$ represent the height of the flagpole. (b) Find the height of the flagpole.

A certificate of deposit with an initial deposit of $\$ 8000$ accumulates $\$ 400$ interest in 2 years. Find the annual interest rate.

You plan to invest $\$ 12,000$ in two funds paying $4 \frac{1}{2} \%$ and $5 \%$ simple interest. (There is more risk in the $5 \%$ fund.) Your goal is to obtain a total annual interest income of $\$ 560$ from the investments. What is the least amount you can invest in the $5 \%$ fund in order to meet your objective?

A grocer mixes peanuts that cost $\$ 2.69$ per pound and walnuts that cost $\$ 4.29$ per pound to make 100 pounds of a mixture that costs $\$ 3.49$ per pound. How much of each kind of nut is put into the mixture?

Forestry A forester mixes gasoline and oil to make 2 gallons of mixture for a two-cycle chainsaw engine. This mixture is 32 parts gasoline and 1 part oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil?

A store has $\$ 40,000$ of inventory in notebook computers and desktop computers. The profit on a notebook computer is $25 \%$ and the profit on a desktop computer is $20 \%$. The profit for the entire stock is $23 \% .$ How much is invested in notebook computers and how much in desktop computers?

A store has $\$ 4500$ of inventory in $8 \times 10$ picture frames and $5 \times 7$ picture frames. The profit on an $8 \times 10$ frame is $25 \%$ and the profit on a $5 \times 7$ frame is $22 \%$. The profit on the entire stock is $24 \% .$ How much is invested in the $8 \times 10$ picture frames and how much is invested in the $5 \times 7$ picture frames?

A triangular sail has an area of 182.25 square feet. The sail has a base of 13.5 feet. Find the height of the sail.

The figure shows three squares. The perimeter of square I is 20 inches and the perimeter of square II is 32 inches. Find the area of square III.

The volume of a rectangular package is 2304 cubic inches. The length of the package is 3 times its width, and the height is $1 \frac{1}{2}$ times its width. (a) Draw a diagram that illustrates the problem. Label the height, width, and length accordingly. (b) Find the dimensions of the package.

The volume of a globe is about 47,712.94 cubic centimeters. Use a graphing utility to find the radius of the globe. Round your result to two decimal places.

The line graph shows the temperatures (in degrees Fahrenheit) on a summer day in Buffalo, New York from 10: 00 A.M. to 6: 00 P.M. Create a new line graph showing the temperatures throughout the day in degrees Celsius.

The average the contiguous United Sta the average temperature in degrees Celsius?

You are driving on a Canadian freeway to a town that is 300 kilometers from your home. After 30 minutes you pass a freeway exit that you know is 50 kilometers from your home. Assuming that you continue at the same constant speed, how long will it take for the entire trip?

A gondola tower in an amusement park casts a shadow that is 80 feet long while a sign that is 4 feet tall casts a shadow that is $3 \frac{1}{2}$ feet long. Draw a diagram for the situation. Then find the height of the tower.

Physics In Exercises 73 and $74,$ you have a uniform beam of length $L$ with a fulcrum $x$ feet from one end (see figure). Objects with weights $W_{1}$ and $W_{2}$ are placed at opposite ends of the beam. The beam will balance when $W_{1} x=W_{2}(L-x)$ Find $x$ such that the beam will balance.Two children weighing 50 pounds $\left(W_{1}\right)$ and 75 pounds $\left(W_{2}\right)$ are going to play on a seesaw that is 10 feet long.

You have a uniform beam of length $L$ with a fulcrum $x$ feet from one end (see figure). Objects with weights $W_{1}$ and $W_{2}$ are placed at opposite ends of the beam. The beam will balance when $W_{1} x=W_{2}(L-x)$ Find $x$ such that the beam will balance. A person weighing 200 pounds $\left(W_{1}\right)$ is attempting to move a 550 -pound rock $\left(W_{2}\right)$ with a bar that is 5 feet long.

Determine whether the statement is true or false. Justify your answer. The equation $x(3-x)=10$ is a linear equation.

Determine whether the statement is true or false. Justify your answer. The volume of a cube with a side length of 9.5 inches is greater than the volume of a sphere with a radius of 5.9 inches.

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-3$$

Write a linear equation that has the given solution. (There are many correct answers.) $$x=0$$

Write a linear equation that has the given solution. (There are many correct answers.) $$x=\frac{1}{4}$$

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-2.5$$

Describe the error in solving the equation. $\frac{1}{x+1}+\frac{1}{x-1}=\frac{2}{(x+1)(x-1)}$ $(x-1)+(x+1)=2$ $2 x=2$ $x=$ The solution is $x=1$

Consider the equation $\frac{6}{(x-3)(x-1)}=\frac{3}{x-3}+\frac{4}{x-1}$ Without performing any calculations, explain how to clear this equation of fractions. Is it possible that this process will introduce an extraneous solution? If so, describe two ways to determine whether a solution is extraneous.

Find $c$ such that $x=3$ is a solution of the linear equation $2 x-5 c=10+3 c-3 x$

Find $c$ such that $x=2$ is a solution of the linear equation $5 x+2 c=12+4 x-2 c$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=\frac{5}{8} x-2$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=\frac{3 x-5}{2}+2$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=(x-3)^{2}+7$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=\frac{1}{3} x^{2}-4$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=-\frac{1}{2}|x+4|-1$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=|x-2|+10$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$(f+g)(-3)$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$(g-f)(-1)$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$(f g)(8)$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$\left(\frac{f}{g}\right)\left(\frac{1}{2}\right)$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$(f \circ g)(4)$$

Evaluate the combination of functions for $f(x)=-x^{2}+4$ and $g(x)=6 x-5$ $$(g \cdot f)(2)$$