Question

question
solve for ( x ) and graph the solution on the number line below.
( -18 leq -x - 6 < -15 )

answer attempt 1 out of 3
inequality notation: (square)
number line:
(number line from -12 to 12 with tick marks at -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12; instruction: click and drag to plot line.)
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Explanation:

Step1: Add 6 to all parts

To isolate the term with \(x\), we add 6 to each part of the compound inequality.
\[
-18 + 6 \leq -x - 6 + 6 < -15 + 6
\]
Simplifying each part:
\[
-12 \leq -x < -9
\]

Step2: Multiply by -1 (reverse inequalities)

When we multiply or divide an inequality by a negative number, the direction of the inequality signs reverses. Multiply each part by \(-1\):
\[
(-1)\times(-12) \geq (-1)\times(-x) > (-1)\times(-9)
\]
Simplifying each part:
\[
12 \geq x > 9
\]
Which can be rewritten as:
\[
9 < x \leq 12
\]

Answer:

Inequality Notation: \(9 < x \leq 12\)

For the number line: Plot an open circle at 9 (since \(x > 9\), not including 9) and a closed circle at 12 (since \(x \leq 12\), including 12), then draw a line segment connecting the two circles.