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Solve for $v$ . $|{-4v}| \leq 4$ Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$ . Use integers, proper fractions, or improper fractions in simplest form.

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Solve absolute value inequalities

Full solution

Q. Solve for

v

.

|{-4v}| \leq 4

Write a compound inequality like

1 < x < 3

or like

x < 1

or

x > 3

. Use integers, proper fractions, or improper fractions in simplest form.

Absolute Value Inequality: We have the inequality: $\newline$ $|-4v| \leq 4$ $\newline$ First, we solve for $|-4v|$ . $\newline$ $|-4v| \leq 4$ means that the absolute value of $-4v$ is less than or equal to $4$ .
Splitting into Two Inequalities: The absolute value inequality $|-4v| \leq 4$ can be split into two separate inequalities because the absolute value of a number is the distance from zero, and it can be either positive or negative. $\newline$ So, we have $-4v \leq 4$ and $-4v \geq -4$ .
Solving $-4v \leq 4$ : Now, we solve the first inequality $-4v \leq 4$ . $\newline$ Divide both sides by $-4$ to isolate $v$ . Remember that dividing by a negative number reverses the inequality sign. $\newline$ $v \geq -1$
Solving $-4v \geq -4$ : Next, we solve the second inequality $-4v \geq -4$ . Again, divide both sides by $-4$ , and reverse the inequality sign. $v \leq 1$
Combining Inequalities: Combining both inequalities, we get the compound inequality: $\newline$ $-1 \leq v \leq 1$ $\newline$ This means that $v$ is greater than or equal to $-1$ and less than or equal to $1$ .

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