Solve for z. |-4z| ≤ 4 Write a compound inequality like 1 3. Use integers, proper fractions, or improper fractions in simplest form. ______

Absolute Value Split: We have the inequality: |-4z| ≤ 4 First, we solve for |-4z|. |-4z| ≤ 4 means that the absolute value of -4z is less than or equal to 4.Solving First Inequality: The absolute value inequality |-4z| ≤ 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. So we have: -4z ≤ 4 and -4z ≥ -4Isolating z: Now we solve each inequality separately. Starting with the first one: -4z ≤ 4 To isolate z, we divide both sides by -4. Remember that dividing by a negative number reverses the inequality sign. z ≥ -1Solving Second Inequality: Now we solve the second inequality: -4z ≥ -4 Again, we divide both sides by -4, and reverse the inequality sign. z ≤ 1Combining Inequalities: Combining both inequalities, we get the compound inequality: -1 ≤ z ≤ 1 This is the solution to the original problem.

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Solve for $z$ . $\newline$ $|-4z| \leq 4$ $\newline$ 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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Algebra 1

Solve absolute value inequalities

Full solution

Q. Solve for

z

\newline

|-4z| \leq 4

\newline

Write a compound inequality like

1 < x < 3

or like

x < 1

x > 3

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

\newline

______

Absolute Value Split: We have the inequality: $\newline$ $|-4z| \leq 4$ $\newline$ First, we solve for $|-4z|$ . $\newline$ $|-4z| \leq 4$ means that the absolute value of $-4z$ is less than or equal to $4$ .
Solving First Inequality: The absolute value inequality $|-4z| \leq 4$ 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. So we have: $\newline$ $-4z \leq 4$ and $-4z \geq -4$
Isolating z: Now we solve each inequality separately. Starting with the first one: $\newline$ $-4z \leq 4$ $\newline$ To isolate z, we divide both sides by $-4$ . Remember that dividing by a negative number reverses the inequality sign. $\newline$ $z \geq -1$
Solving Second Inequality: Now we solve the second inequality: $\newline$ $-4z \geq -4$ $\newline$ Again, we divide both sides by $-4$ , and reverse the inequality sign. $\newline$ $z \leq 1$
Combining Inequalities: Combining both inequalities, we get the compound inequality: $\newline$ $-1 \leq z \leq 1$ $\newline$ This is the solution to the original problem.