# 6.1: Inequalities Using Addition and Subtraction

Difficulty Level: At Grade Created by: CK-12

Verbs that translate into inequalities are:

\begin{align*}>\end{align*} “greater than”

\begin{align*}\ge\end{align*} “greater than or equal to”

\begin{align*} < \end{align*} “less than”

\begin{align*}\le\end{align*} “less than or equal to”

\begin{align*}\neq\end{align*} “not equal to”

Definition: An algebraic inequality is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.

Solutions to one-variable inequalities can be graphed on a number line or in a coordinate plane.

Example: Graph the solutions to \begin{align*}t > 3\end{align*} on a number line.

Solution: The inequality is asking for all real numbers larger than 3.

You can also write inequalities given a number line of solutions.

Example: Write the inequality pictured below.

Solution: The value of four is colored in, meaning that four is a solution to the inequality. The red arrow indicates values less than four. Therefore, the inequality is:

\begin{align*}x \le 4\end{align*}

Inequalities that “include” the value are shown as \begin{align*}\le\end{align*} or \begin{align*}\ge\end{align*}. The line underneath the inequality stands for “or equal to.” We show this relationship by coloring in the circle above this value on the number line, as in the previous example. For inequalities without the “or equal to,” the circle above the value on the number line remains unfilled.

## Four Ways to Express Solutions to Inequalities

1. Inequality notation: The answer is expressed as an algebraic inequality, such as \begin{align*}d \le \frac{1}{2}\end{align*}.

2. Set notation: The inequality is rewritten using set notation brackets { }. For example, \begin{align*}\left \{ d | d \le \frac{1}{2} \right \}\end{align*} is read, “The set of all values of \begin{align*}d\end{align*}, such that \begin{align*}d\end{align*} is a real number less than or equal to one-half.”

3. Interval notation: This notation uses brackets to denote the range of values in an inequality.

1. Square or “closed” brackets [ ] indicate that the number is included in the solution
2. Round or “open” brackets ( ) indicate that the number is not included in the solution.

Interval notation also uses the concept of infinity \begin{align*}\infty\end{align*} and negative infinity \begin{align*}- \infty\end{align*}. For example, for all values of \begin{align*}d\end{align*} that are less than or equal to \begin{align*}\frac{1}{2}\end{align*}, you could use set notation as follows: \begin{align*} \left . \left (-\infty, \frac{1}{2}\right \} \right .\end{align*}.

4. As a graphed sentence on a number line.

Example: (8, 24) states that the solution is all numbers between 8 and 24 but does not include the numbers 8 and 24.

[3, 12) states that the solution is all numbers between 3 and 12, including 3 but not including 12.

## Inequalities Using Addition or Subtraction

To solve inequalities, you need some properties.

Addition Property of Inequality: For all real numbers \begin{align*}a, \ b,\end{align*} and \begin{align*}c\end{align*}:

If \begin{align*}x < a\end{align*}, then \begin{align*}x + b < a + b\end{align*}.

If \begin{align*}x < a\end{align*}, then \begin{align*}x - c < a - c\end{align*}.

The two properties above are also true for \begin{align*}\le\end{align*} or \begin{align*}\ge\end{align*}.

Because subtraction can also be thought of as “add the opposite,” these properties also work for subtraction situations.

Just like one-step equations, the goal is to isolate the variable, meaning to get the variable alone on one side of the inequality symbol. To do this, you will cancel the operations using inverses.

Example: Solve for \begin{align*}x:\ x - 3 < 10\end{align*}.

Solution: To isolate the variable \begin{align*}x\end{align*}, you must cancel “subtract 3” using its inverse operation, addition.

\begin{align*}x-3+3 &< 10 + 3\\ x &< 13\end{align*}

Now, check your answer. Choose a number less than 13 and substitute it into your original inequality. If you choose 0, and substitute it you get:

\begin{align*}0 - 3 < 10 = -3 < 10\end{align*}

What happens at 13? What happens with numbers greater than 13?

Example: Solve for \begin{align*}x: \ x + 4 > 13\end{align*}

Solution:

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

## Writing Real-Life Inequalities

As described in the chapter opener, inequalities appear frequently in real life. Solving inequalities is an important part of algebra.

Example: Write the following statement as an algebraic inequality. You must maintain a balance of at least $2,500 in your checking account to avoid a finance charge. Solution: The key phrase in this statement is “at least.” This means you can have$2,500 or more in your account to avoid a finance charge.

Choose the variable to describe the money in your account, say \begin{align*}m\end{align*}.

Write the inequality: \begin{align*}m \ge 2500\end{align*}.

Graph the solutions using a number line.

Example: Translate into an algebraic inequality: “The speed limit is 65 miles per hour.”

Solution: To avoid a ticket, you must drive 65 or less. Choose a variable to describe your possible speed, say \begin{align*}s.\end{align*}

Write the inequality \begin{align*}s \le 65.\end{align*}

Graph the solutions to the inequality using a number line.

In theory, you cannot drive a negative number of miles per hour. This concept will be a focus later in this chapter.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.

1. What are the four methods of writing the solution to an inequality?

Graph the solutions to the following inequalities using a number line.

1. \begin{align*}x < -3\end{align*}
2. \begin{align*}x \ge 6\end{align*}
3. \begin{align*}x > 0\end{align*}
4. \begin{align*}x \le 8\end{align*}
5. \begin{align*}x < -35\end{align*}
6. \begin{align*}x > -17\end{align*}
7. \begin{align*}x \ge 20\end{align*}
8. \begin{align*}x \le 3 \end{align*}

Write the inequality that is represented by each graph.

Write the inequality given by the statement. Choose an appropriate letter to describe the unknown quantity.

1. You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.
2. You must be younger than 3 years old to get free admission at the San Diego Zoo.
3. Charlie needs more than \$1,800 to purchase a car.
4. Cheryl can have no more than six pets at her house.
5. The shelter can house no more than 16 rabbits.

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

1. \begin{align*}x - 1 > -10\end{align*}
2. \begin{align*}x - 1 \le -5\end{align*}
3. \begin{align*}-20 + a \ge 14\end{align*}
4. \begin{align*}x + 2 < 7\end{align*}
5. \begin{align*}x + 8 \le -7\end{align*}
6. \begin{align*}5 + t \ge \frac{3}{4}\end{align*}
7. \begin{align*}x - 5 < 35\end{align*}
8. \begin{align*}15 + g \ge -60\end{align*}
9. \begin{align*}x - 2 \le 1\end{align*}
10. \begin{align*}x - 8 > -20\end{align*}
11. \begin{align*}11 + q > 13\end{align*}
12. \begin{align*}x + 65 < 100\end{align*}
13. \begin{align*}x - 32 \le 0\end{align*}
14. \begin{align*}x + 68 \ge 75\end{align*}
15. \begin{align*}16 + y \le 0\end{align*}

Mixed Review

1. Write an equation containing (3, –6) and (–2, –2).
2. Simplify: \begin{align*}|2-11 \times 3| + 1\end{align*}.
3. Graph \begin{align*}y=-5\end{align*} on a coordinate plane.
4. \begin{align*}y\end{align*} varies directly as \begin{align*}x\end{align*}. When \begin{align*}x=-1, \ y = \frac{4}{5}\end{align*}. Find \begin{align*}y\end{align*} when \begin{align*}x = \frac{16}{3}\end{align*}.
5. Rewrite in slope-intercept form: \begin{align*}-2x\end{align*}.
6. \begin{align*}-2x + 7y = 63\end{align*}

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