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6.1: Inequalities Using Addition and Subtraction

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Write and graph inequalities in one variable on a number line.
  • Solve an inequality using addition.
  • Solve an inequality using subtraction.

Introduction

Inequalities are similar to equations in that they show a relationship between two expressions. We solve and graph inequalities in a similar way to equations. However, there are some differences that we will talk about in this chapter. The main difference is that for linear inequalities the answer is an interval of values whereas for a linear equation the answer is most often just one value.

When writing inequalities we use the following symbols

\begin{align*}> \qquad \text{is greater than}\!\\ \geq \qquad \text{is greater than or equal to}\!\\ < \qquad \text{is less than}\!\\ \leq \qquad \text{is less than or equal to}\end{align*}

Write and Graph Inequalities in One Variable on a Number Line

Let’s start with the simple inequality \begin{align*}x>3\end{align*}

We read this inequality as “\begin{align*}x\end{align*} is greater than 3”. The solution is the set of all real numbers that are greater than three. We often represent the solution set of an inequality by a number line graph.

Consider another simple inequality \begin{align*}x \leq 4\end{align*}

We read this inequality as “\begin{align*}x\end{align*} is less than or equal to 4”. The solution is the set of all real numbers that equal four or less than four. We graph this solution set on the number line.

In a graph, we use an empty circle for the endpoint of a strict inequality \begin{align*}(x>3)\end{align*} and a filled circle if the equal sign is included \begin{align*}(x\leq 4)\end{align*}.

Example 1

Graph the following inequalities on the number line.

a) \begin{align*}x < -3\end{align*}

b) \begin{align*}x \geq 6\end{align*}

c) \begin{align*}x > 0\end{align*}

d) \begin{align*}x \leq 8\end{align*}

Solution

a) The inequality \begin{align*}x < -3\end{align*} represents all real numbers that are less than -3. The number -3 is not included in the solution and that is represented by an open circle on the graph.

b) The inequality \begin{align*}x \geq 6\end{align*} represents all real numbers that are greater than or equal to six. The number six is included in the solution and that is represented by a closed circle on the graph.

c) The inequality \begin{align*}x > 0\end{align*} represents all real numbers that are greater than zero. The number zero is not included in the solution and that is represented by an open circle on the graph.

d) The inequality \begin{align*}x \leq 8\end{align*} represents all real numbers that are less than or equal to eight. The number eight is included in the solution and that is represented by a closed circle on the graph.

Example 2

Write the inequality that is represented by each graph.

a)

b)

c)

d)

Solution:

a) \begin{align*}x \leq -12\end{align*}

b) \begin{align*} x > 540 \end{align*}

c) \begin{align*}x < 65\end{align*}

d) \begin{align*} x \geq 85\end{align*}

Inequalities appear everywhere in real life. Here are some simple examples of real-world applications.

Example 3

Write each statement as an inequality and graph it on the number line.

a) You must maintain a balance of at least $2500 in your checking account to get free checking.

b) You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.

c) You must be younger than 3 years old to get free admission at the San Diego Zoo.

d) The speed limit on the interstate is 65 miles per hour.

Solution:

a) The inequality is written as \begin{align*}x \geq 2500\end{align*}. The words “at least” imply that the value of $2500 is included in the solution set.

b) The inequality is written as \begin{align*} x \geq 48\end{align*}. The words “at least” imply that the value of 48 inches is included in the solution set.

c) The inequality is written as \begin{align*}x < 3\end{align*}.

d) Speed limit means the highest allowable speed, so the inequality is written as \begin{align*}x \leq 65\end{align*}.

Solve an Inequality Using Addition

To solve an inequality we must isolate the variable on one side of the inequality sign. To isolate the variable, we use the same basic techniques used in solving equations. For inequalities of this type:

\begin{align*}x-a <b\end{align*} or \begin{align*}x-a > b\end{align*}

We isolate the \begin{align*}x\end{align*} by adding the constant \begin{align*}a\end{align*} to both sides of the inequality.

Example 4

Solve each inequality and graph the solution set.

a) \begin{align*}x-3 <10\end{align*}

b) \begin{align*}x-1 >-10\end{align*}

c) \begin{align*}x-1 \leq -5\end{align*}

d) \begin{align*}x-20 \geq 14\end{align*}

Solution:

a) \begin{align*}\text{To solve the inequality} & & x-3 &<10\\ \text{Add} \ 3 \ \text{to both sides of the inequality}. & & x-3+3 &<10+3\\ \text{Simplify} & & x &<13\end{align*}

b) \begin{align*}\text{To solve the inequality} & & x-1 & >-10\\ \text{Add} \ 12 \ \text{to both sides of the inequality} & & x-12+12 & >-10+12\\ \text{Simplify} & & x & >2\end{align*}

c) \begin{align*}\text{To solve the inequality} & & x-1 & \leq -5\\ \text{Add} \ 1 \ \text{to both sides of the inequality} & & x-1+1 & \leq -5+1\\ \text{Simplify to obtain} & & x & \leq -4\end{align*}

d) \begin{align*}\text{To solve the inequality} & & x-20 & \leq 14\\ \text{Add} \ 20 \ \text{to both sides of the inequality}: & & x-20+20 & \leq 14+20\\ \text{Simplify} & & x & \leq 34\end{align*}

Solve an Inequality Using Subtraction

For inequalities of this type:

\begin{align*}x+1<b\end{align*} or \begin{align*}x+1 > b\end{align*}

We isolate the \begin{align*}x\end{align*} by subtracting the constant \begin{align*}a\end{align*} on both sides of the inequality.

Example 5

Solve each inequality and graph the solution set.

a) \begin{align*}x+2 <7\end{align*}

b) \begin{align*}x+8 \leq -7\end{align*}

c) \begin{align*}x+4 >13\end{align*}

d) \begin{align*}x+5 \geq - \frac{3}{4}\end{align*}

Solution:

a) \begin{align*}\text{To solve the inequality} & & x+2 &< 7\\ \text{Subtract} \ 2 \ \text{on both sides of the inequality} & & x+2-2 &<7-2\\ \text{Simplify to obtain} & & x&<5\end{align*}

b) \begin{align*}\text{To solve the inequality} & & x+8 &\leq -7\\ \text{Subtract} \ 8 \ \text{on both sides of the inequality} & & x+8-8 & \leq -7-8\\ \text{Simplify to obtain}: & & x & \leq -15\end{align*}

c) \begin{align*}\text{To solve the inequality} & & x+4 & >13\\ \text{Subtract} \ 4 \ \text{on both sides of the inequality} & & x + 4 -4 & > 13 - 4\\ \text{Simplify} & & x & >9\end{align*}

d) \begin{align*}\text{To solve the inequality} & & x + 5 & \geq \frac{3} {4}\\ \text{Subtract} \ 5 \ \text{on both sides of the inequality} & & x + 5 -5 & \geq -\frac{3} {4}-5\\ \text{Simplify to obtain}: & & x & \geq -5\frac{3} {4}\end{align*}

Lesson Summary

  • The answer to an inequality is often an interval of values. Common inequalities are:
  • > is greater than
  • \begin{align*} \geq\end{align*} is greater than or equal to
  • > is less than
  • \begin{align*} \leq \end{align*} is less than or equal to
  • Solving inequalities with addition and subtraction works just like solving an equation. To solve, we isolate the variable on one side of the equation.

Review Questions

  1. Write the inequality represented by the graph.
  2. Write the inequality represented by the graph.
  3. Write the inequality represented by the graph.
  4. Write the inequality represented by the graph.

Graph each inequality on the number line.

  1. \begin{align*} x< -35\end{align*}
  2. \begin{align*} x > -17\end{align*}
  3. \begin{align*} x \geq 20\end{align*}
  4. \begin{align*} x \leq 3 \end{align*}

Solve each inequality and graph the solution on the number line.

  1. \begin{align*} x-5 < 35\end{align*}
  2. \begin{align*} x+ 15 \geq -60\end{align*}
  3. \begin{align*} x-2 \leq 1\end{align*}
  4. \begin{align*} x-8 > -20\end{align*}
  5. \begin{align*} x+11 > 13\end{align*}
  6. \begin{align*} x+ 65 < 100 \end{align*}
  7. \begin{align*} x-32 \leq 0\end{align*}
  8. \begin{align*} x+68 \geq 75\end{align*}

Review Answers

  1. \begin{align*} x \geq 1\end{align*}
  2. \begin{align*} x <-10 \end{align*}
  3. \begin{align*} x \leq -10\end{align*}
  4. \begin{align*}x > 30\end{align*}

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