Exercise 9 Page 331

First, let's split the compound inequality into separate inequalities. Compound Inequality: -4< 3x&+ 8 ≤ 23 First Inequality: -4< 3x &+8 Second Inequality: 3x&+ 8 ≤ 23 Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and. -4< 3x+8 and 3x+ 8 ≤ 23

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<3x+8

SubIneq

LHS-8

-12<3x

DivIneq

.LHS /3.<.RHS /3.

-4

RearrangeIneq

Rearrange inequality

x>-4

The first inequality is satisfied by all values greater than -4. Notice that x cannot be -4 as the inequality sign is strict.

Second inequality

Once more, we will solve the inequality by isolating the variable.

3x+8≤23

SubIneq

LHS-8≤RHS-8

3x≤15

DivIneq

.LHS /3.≤.RHS /3.

x≤5

The second inequality is true for numbers less than or equal to 5.

Compound inequality

The solution set to the compound inequality is the intersection of the solution sets. First Solution Set: -4 < x& Second Solution Set: x&≤5 Intersecting Solution Set: -4 < x&≤5