Question

For the following exercises, solve each inequality and write the solution in interval notation.

30. 2| v − 7 | − 4 ≥ 42

Answer 1

The solution to the inequality
`$2|v-7|-4 \geq 42$` in interval notation is given by
`$-16 \leq v \leq 30$`.

Explanation:

An absolute value inequality
`$2|v-7|-4 \geq 42$` is given.

It is required to solve the inequality and write the solution in interval form.

To write the solution, first solve the given absolute value inequality algebraically and then write it in interval notation.

Step 1 of 4

The given absolute value inequality is
`$2|v-7|-4 \geq 42$`.

Add on both 4 sides,

`$$\begin{aligned}&2|v-7|-4 \geq 42 \\&2|v-7|-4+4 \geq 42+4 \\&2|v-7| \geq 46\end{aligned}$$`

Step 2 of 4

Divide by 2 on both sides,

`$$\begin{aligned}&(2|v-7|)/(2) \geq (46)/(2) \\&|v-7| \geq 23\end{aligned}$$`

The inequality can be written as
`$v-7 \leq 23$` and
`$v-7 \geq-23$`

Step 3 of 4

First solve the inequality,
`$v-7 \leq 23$`.

Add 7 on both sides,

`$$\begin{aligned}&v-7 \leq 23 \\&v-7+7 \leq 23+7 \\&v \leq 30\end{aligned}$$`

Step 4 of 4

Solve the inequality
`$v-7 \geq-23$`.

Add 7 on both sides,

`$$\begin{aligned}&v-7 \geq-23 \\&v-7+7 \geq-23+7 \\&v \geq-16\end{aligned}$$`

The solution of the inequality in interval notation is given by
`$-16 \leq v \leq 30$`.