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Solve for $z$ . $|4z| > 4$ 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

z

.

|4z| > 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.

Isolate absolute value: We have the inequality $|4z| > 4$ . To solve for $z$ , we first isolate the absolute value expression on one side of the inequality. $\newline$ $|4z| > 4$
Recognize absolute value inequality: Next, we recognize that the absolute value inequality $|4z| > 4$ means that the expression inside the absolute value, $4z$ , is either greater than $4$ or less than $-4$ . This is because the absolute value of a number is its distance from zero on the number line, so if the distance is greater than $4$ , the number itself could be either greater than $4$ or less than $-4$ . So we can write two separate inequalities: $4z > 4$ or $4z < -4$
Solve first inequality: Now we solve each inequality for $z$ . Starting with the first inequality: $\newline$ $4z > 4$ $\newline$ Divide both sides by $4$ to isolate $z$ : $\newline$ $z > 1$
Solve second inequality: Now we solve the second inequality: $\newline$ $4z < -4$ $\newline$ Divide both sides by $4$ to isolate $z$ : $\newline$ $z < -1$
Combine solutions: Combining the two solutions, we get the compound inequality for $z$ : $z > 1$ or $z < -1$ This is the solution to the original inequality $|4z| > 4$ .

More problems from Solve absolute value inequalities

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Which of the following best describes the solutions to the inequality shown?

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1

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