Chapter 8.7, Problem 20IP

To determine

Solve the inequality and check the solutions.

Answer to Problem 20IP

The solution of $13\ge 9+b$ is $b\le 4\text{or}(-\infty ,4]$

Explanation of Solution

Given:

The Inequality: $13\ge 9+b$

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:

The inequality: $13\ge 9+b$

  $\begin{array}{l}13\ge 9+b\\ 13-9\ge 9-9+b\text{[}\text{Subtract}9\text{from}\text{both}\text{sides}\text{]}\\ 4\ge b\\ b\le 4[b\ge c\Rightarrow c\le b]\end{array}$

  $\begin{array}{l}\text{Check}\text{the}\text{solution}\text{when}b=0\\ 13\ge 9+b\\ 13\ge 9+0\\ 13\ge 9[\text{True}\text{Statement}]\end{array}$

The solution of $13\ge 9+b$ is $b\le 4\text{or}(-\infty ,4]$

Thus, the solution of $13\ge 9+b$ is $b\le 4\text{or}(-\infty ,4]$