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Solve for $u$ . $|u - 3| \geq 17$ 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

u

.

|u - 3| \geq 17

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

______

Absolute Value Inequality: We have the inequality $|u - 3| \geq 17$ . The absolute value inequality means that the expression inside the absolute value, $u - 3$ , can either be greater than or equal to $17$ or less than or equal to $-17$ .
Case $1$ : Greater Than or Equal to $17$ : First, let's solve for the case where $u - 3$ is greater than or equal to $17$ : $u - 3 \geq 17$ Add $3$ to both sides to isolate $u$ : $u \geq 17 + 3$ $u \geq 20$
Case $2$ : Less Than or Equal to $-17$ : Now, let's solve for the case where $u - 3$ is less than or equal to $-17$ : $u - 3 \leq -17$ Add $3$ to both sides to isolate $u$ : $u \leq -17 + 3$ $u \leq -14$
Combining Cases: Combining both cases into a compound inequality, we get: $\newline$ $u \leq -14$ or $u \geq 20$ $\newline$ This is the solution to the inequality $|u - 3| \geq 17$ .

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