# SOLVING AND INTERPRETING SOLUTIONS OF INEQUALITIES

In math, we solve inequalities and give solution in different ways like interval notation and graphing.

To have a clear understanding of the problem, we may have to interpret the part of the solution or sometimes the whole.

Let us see, how solution of an inequality can be interpreted through the following examples.

## Examples

Example 1 :

Serena wants to complete the first 3 miles of a 10-mile run in 45 minutes or less running at a steady pace. The inequality 10 - 0.75p ≤ 7 can be used to find p, the pace, in miles per hour, she can run to reach her goal. Solve the inequality. Then graph and interpret the solution.

Solution :

Step 1 :

Use inverse operations to solve the inequality.

10 - 0.75p ≤ 7

Subtract 10 from both sides.

(10 - 0.75p) - 10 ≤ (7) - 10

-0.75p ≤ -3

Divide both sides by -0.75.

(-0.75p) / (-0.75)  (-3) / (-0.75)

≥ 4

Step 2 :

Graph the inequality and interpret the circle and the arrow.

So, Serena has to run at a steady pace of at least 4 miles per hour.

Example 2 :

Joshua wants to complete the first mile of a 5-mile run in 10 minutes or less running at a steady pace. The inequality 5 - p/6 ≤ 4 can be used to find p, the pace, in miles per hour, he can run to reach his goal. Solve the inequality. Then graph and interpret the solution.

Solution :

Step 1 :

Use inverse operations to solve the inequality.

5 - p/6 ≤ 4

Subtract 5 from both sides.

(5 - p/6) - 5 ≤ (4) - 5

- p/6 ≤ -1

Multiply both sides by -6.

(-p/6).(-6)  (-1).(-6)

≥ 6

Step 2 :

Graph the inequality and interpret the solution.

So, Joshua has to run at a steady pace of at least 6 mi/h.

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