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Solve for $z$ . $|z| - 2 \leq 5$ 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

.

|z| - 2 \leq 5

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 start with the given absolute value inequality: $\newline$ $|z| - 2 \leq 5$ $\newline$ First, we isolate the absolute value expression by adding $2$ to both sides of the inequality. $\newline$ $|z| - 2 + 2 \leq 5 + 2$ $\newline$ $|z| \leq 7$
Consider absolute value definition: Now we need to consider the definition of absolute value, which states that |z|\(\newline) is the distance of z\(\newline) from \(0 $\newline$ ) on the number line. The inequality |z| \leq \(7 $\newline$ ) means that z\(\newline) is within a distance of \(7 $\newline$ ) from \(0 $\newline$ ). This gives us two scenarios: $\newline$ $1$ . z \leq \(7 $\newline$ ). $\newline$ $2$ . z \geq \(-7 $\newline$ ).
Write compound inequality: We can write these two scenarios as a compound inequality: $\newline$ $-7 \leq z \leq 7$ $\newline$ This compound inequality represents all the values of $z$ that satisfy the original inequality $|z| - 2 \leq 5$ .

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