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Solve for $z$ . $\newline$ $|-2z| \leq 18$ $\newline$ 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. $\newline$ ______

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Solve absolute value inequalities

Full solution

Q. Solve for

z

.

\newline

|-2z| \leq 18

\newline

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.

\newline

______

Given Inequality: We are given the inequality: $\newline$ $|-2z| \leq 18$ $\newline$ First, we need to solve for $|2z|$ . $\newline$ $|-2z| \leq 18$ $\newline$ This means that the absolute value of $-2z$ is less than or equal to $18$ .
Split Absolute Value: The absolute value inequality $|-2z| \leq 18$ 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. Therefore, we have: $\newline$ $-2z \leq 18$ and $-2z \geq -18$
Solve First Inequality: Now we solve each inequality for $z$ . Starting with the first inequality: $\newline$ $-2z \leq 18$ $\newline$ Divide both sides by $-2$ to isolate $z$ . Remember that dividing by a negative number reverses the inequality sign. $\newline$ $z \geq -9$
Solve Second Inequality: Now we solve the second inequality: $\newline$ $-2z \geq -18$ $\newline$ Divide both sides by $-2$ , again remembering to reverse the inequality sign. $\newline$ $z \leq 9$
Combine Inequalities: Combining both inequalities, we get the compound inequality: $\newline$ $-9 \leq z \leq 9$ $\newline$ This is the solution to the original problem.

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