Exercise 15 Page 229

We were asked to solve a compound inequality. Let's start by splitting it into separate inequalities. Compound Inequality:&& 4< 6b&-2≤ 28 First Inequality:&& 4< 6b&-2 Second Inequality:&& 6b&-2≤ 28 Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and. 4< 6b-2 and 6b-2≤ 28

Let's solve the inequalities separately.

First Inequality

Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must flip the inequality sign.

4<6b-2

AddIneq

LHS+2

6<6b

DivIneq

.LHS /6.<.RHS /6.

1

RearrangeIneq

Rearrange inequality

b>1

The above tells us that all values greater than 1 will satisfy the first inequality.

Second inequality

Now we can solve the second inequality.

6b-2≤ 28

AddIneq

LHS+2≤RHS+2

6b≤ 30

DivIneq

.LHS /6.≤.RHS /6.

b≤ 5

The above tells us that all values less than or equal to 5 will satisfy the second inequality.

Combining solution sets

The solution set to the compound inequality is the intersection of the solution sets. To make things easier we will rewrite b>1 as 1D.