Question

solve the linear inequality then, a graph the solution on a number line, b write the solution in set - builder notation, and c write the solution in interval notation (\frac{18 - 11w}{8}-\frac{w}{8}<-12)

b select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.

a. the solution is ({w|w < square})
(simplify your answer use integers or fractions for any numbers in the inequality.)

b. the solution is ({w|square < w < square})
(simplify your answers use integers or fractions for any numbers in the inequality.)

c. the solution is ({w|w > square})
(simplify your answer use integers or fractions for any numbers in the inequality.)

d. the solution is the empty set

c select the correct choice and, if necessary, fill in the answer box to complete your choice

a. the solution is (square)
(type your answer in interval notation simplify your answer use integers or fractions for any numbers in the expression.)

b. the solution is the empty set

Explanation:

Step1: Combine the fractions

Since both terms have a denominator of 8, we can combine them: $\frac{18 - 11w - w}{8} < -12$
Simplify the numerator: $\frac{18 - 12w}{8} < -12$

Step2: Multiply both sides by 8

To eliminate the denominator, multiply both sides by 8: $18 - 12w < -12 \times 8$
Calculate $-12 \times 8$: $18 - 12w < -96$

Step3: Subtract 18 from both sides

Subtract 18 from each side: $-12w < -96 - 18$
Simplify the right side: $-12w < -114$

Step4: Divide by -12 (and reverse inequality)

Divide both sides by -12. Remember to reverse the inequality sign: $w > \frac{-114}{-12}$
Simplify the fraction: $w > \frac{19}{2}$ (or 9.5)

Part b

The solution in set - builder notation is for values of $w$ greater than $\frac{19}{2}$. So the correct choice is C. The solution is $\{w|w > \frac{19}{2}\}$

Part c

In interval notation, the set of all real numbers greater than $\frac{19}{2}$ is written as $(\frac{19}{2}, \infty)$. So the correct choice is A. The solution is $(\frac{19}{2}, \infty)$

For part b:

Answer:

C. The solution is $\{w|w > \frac{19}{2}\}$

For part c: