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Solve for $y$ . $|y - 3| \geq 12$ 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

y

.

|y - 3| \geq 12

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 $|y - 3| \geq 12$ . The absolute value inequality means that the expression inside the absolute value can be either greater than or equal to $12$ or less than or equal to $-12$ .
Case $1$ : Greater Than or Equal to $12$ : First, we consider the case where the expression inside the absolute value is greater than or equal to $12$ : $y - 3 \geq 12$ Now, we solve for $y$ by adding $3$ to both sides of the inequality: $y - 3 + 3 \geq 12 + 3$ $y \geq 15$
Case $2$ : Less Than or Equal to $-12$ : Next, we consider the case where the expression inside the absolute value is less than or equal to $-12$ : $\newline$ $y - 3 \leq -12$ $\newline$ Again, we solve for $y$ by adding $3$ to both sides of the inequality: $\newline$ $y - 3 + 3 \leq -12 + 3$ $\newline$ $y \leq -9$
Compound Inequality: Combining both cases, we get a compound inequality: $\newline$ $y \leq -9$ or $y \geq 15$ $\newline$ This is the solution to the given inequality.

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