Chapter 8.6, Problem 21IP

To determine

Find for the given value, state whether the inequality $\frac{2}{5}c+18\le 25;c=15$ is true or false.

Answer to Problem 21IP

True

Explanation of Solution

Given:

The Inequality: $\frac{2}{5}c+18\le 25;c=15$

Concept Used:

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. a ≠ b says that a is not equal to b. a < b says that a is less than b. a > b says that a is greater than b.

There are four different types of inequalities:

Greater than − $(>)$ ; Less than − $(<)$ ; Greater than or equal to − $(\ge )$ ; Less than or equal to − $(\le )$

For Inequality equation: If $b>c\Rightarrow c<b\text{or}b<c\Rightarrow c>b$

Rules for solving inequality equations:

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

But these things do change the direction of the inequality (" $<$ " becomes " $>$ " for example):

• Multiply (or divide) both sides by a negative number
• Swapping left and right hand sides
• If there is a parenthesis $\left(\right)$ on the interval, then we use the strict inequality symbols $<\text{ or}>$

If there is a bracket $\left[\right]$ on the interval, then we use the inequality symbols

Calculation:

  $\begin{array}{l}\frac{2}{5}c+18\le 25\\ \frac{2}{5}c+18-18\le 25-18\text{[}\text{Subtract}18\text{from}\text{both}\text{sides}\text{]}\\ \frac{2}{5}c\le 7\\ \frac{5}{2}·\frac{2}{5}c\le \frac{5}{2}·7\text{[}\text{Multiply}\frac{5}{2}\text{both}\text{sides}\text{by}\text{]}\\ c\le 17.5\end{array}$

Solution $c\le 17.5$ means the number less than or equal to 17.5.

  $c=15$ is less than 17, so the value $c=15$ is true.

  

Thus, it means $c=15$ is true. True