REVIEW: Solving Linear Inequalities

Previous Material: Solving Linear Inequalities Part 1

READ: Solve Inequalities and Graph Solutions

Solve Inequalities and Graph Solutions

Now that you have developed an understanding of inequalities we can solve them and graph the solution sets.

We can solve an inequality in a similar way as we would use to solve an equation. The tricky part is in the answer not in the process. Once we solve the inequality, we can graph the solution. Let’s look at an example.


Example 1

Solve this inequality and graph its solution n-4 \le 3.

Solve the inequality as you would solve an equation, by using inverse operations. Since the 4 is subtracted from n, add 4 to both sides of the inequality to solve it. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol. The symbol should stay the same.

n-4 & \le 3\\ n-4 +4 & \le 3+4\\ n+(-4+4) & \le 7\\ n+0 & \le 7\\ n & \le 7

Now, graph the solution. The inequality n \le 7 is read as “n is less than or equal to 7.” So, the solutions of this inequality include 7 and all numbers that are less than 7.

Draw a number line from 0 to 10. Add a closed circle at 7 to show that 7 is a solution for this inequality. Then draw an arrow showing all numbers less than 7.

The solution for this inequality is n \le 7, and its graph is shown above.


Example 2

Solve this inequality and graph its solution -2n<14.

Solve this inequality as you would solve an equation, by using inverse operations. Since the is multiplied by the n, divide both sides of the inequality by to solve it. Since this involves multiplying both sides of the inequality by a negative number, the sense of the inequality will change and you will need to reverse the inequality symbol. This means changing the inequality symbol from a “less than” symbol (<) to a “greater than” symbol (>).

-2n & < 14\\ \frac{-2n}{-2} & > \frac{14}{-2}\\ 1n & > -7\\ n & > -7

Now, graph the solution. The inequality n>-7 is read as “n is greater than or equal to 7.” So, the solutions of this inequality include all numbers that are greater than 7.

Draw a number line from –10 to 0. Add an open circle at -7 to show that -7 is not a solution for this inequality. Then draw an arrow showing all numbers greater than -7.

The solution for this inequality is n>-7, and its graph is shown above.


Sometimes, you will need to take more than one step to solve an inequality. You can think of these problems in the same way that you thought about two-step equations.


Example 3

\frac{n}{3}+9 \ge -9.

Solve this inequality as you would solve an equation, by using inverse operations. First, try to get the term with the variable, \frac{n}{3}, by itself on one side of the inequality. Since the 9 is being added to \frac{n}{3}, subtract 9 from both sides of the inequality. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol during this step.

\frac{n}{3}+9 & \ge -9\\ \frac{n}{3}+9-9 & \ge -9-9\\ \frac{n}{3}+0 & \ge (-9+-9)\\ \frac{n}{3} & \ge -18

There is a second step you must take to find the solution. Since n is divided by 3, you must multiply both sides of the inequality by 3 to find its solution. This involves multiplying by a positive number, 3, so you do not need to reverse the inequality symbol. Be careful! It is true that you will need to multiply 3 by –18 to find the solution. However, since you are not multiplying both sides of the inequality by a negative number, you do not reverse the inequality symbol.

\frac{n}{3} & \ge -18\\ \frac{n}{3} \times 3 & \ge -18 \times 3\\ \frac{n}{3} \times \frac{3}{1} & \ge -54\\ \frac{n}{\cancel{3}} \times \frac{\cancel{3}}{1} & \ge -54\\ \frac{n}{1} & \ge -54\\ n & \ge -54

The solution for this inequality is n \ge -54.