Question

select the correct answer.
which graph represents the solution to this inequality?
$-\frac{1}{4}(12x + 8) \leq -2x + 11$
\\(\bigcirc\\) a. \\(\leftarrow\frac{}{-17}\frac{}{-15}\frac{}{-13}\frac{}{-11}\frac{}{-9}\frac{}{-7}\frac{}{-5}\
ightarrow\\) with a red dot at -9
\\(\bigcirc\\) b. \\(\leftarrow\frac{}{-17}\frac{}{-15}\frac{}{-13}\frac{}{-11}\frac{}{-9}\frac{}{-7}\frac{}{-5}\
ightarrow\\) with a red dot at -13 and the arrow pointing left
\\(\bigcirc\\) c. \\(\leftarrow\frac{}{-17}\frac{}{-15}\frac{}{-13}\frac{}{-11}\frac{}{-9}\frac{}{-7}\frac{}{-5}\
ightarrow\\) with a red dot at -13 and the arrow pointing right
\\(\bigcirc\\) d. \\(\leftarrow\frac{}{-17}\frac{}{-15}\frac{}{-13}\frac{}{-11}\frac{}{-9}\frac{}{-7}\frac{}{-5}\
ightarrow\\) with a red dot at -9 and the arrow pointing left

Explanation:

Step1: Simplify the left side

Multiply out the left - hand side of the inequality \(-\frac{1}{4}(12x + 8)\leq-2x + 11\). Using the distributive property \(a(b + c)=ab+ac\), where \(a =-\frac{1}{4}\), \(b = 12x\) and \(c = 8\), we get:
\(-\frac{1}{4}\times12x-\frac{1}{4}\times8\leq-2x + 11\)
\(- 3x-2\leq-2x + 11\)

Step2: Isolate the variable x

Add \(3x\) to both sides of the inequality to get:
\(-3x-2 + 3x\leq-2x + 11+3x\)
Simplifying both sides, we have \(-2\leq x + 11\)
Then subtract 11 from both sides:
\(-2-11\leq x+11 - 11\)
\(-13\leq x\), which is equivalent to \(x\geq - 13\)

Step3: Analyze the number line representation

The inequality \(x\geq - 13\) means that we have a closed circle (because the inequality is non - strict, \(\geq\)) at \(x=-13\) and the arrow points to the right (since \(x\) is greater than or equal to \(-13\)). Looking at the options, option C has a closed circle at \(-13\) and the arrow pointing to the right.

Answer:

C. The number line with a closed circle at -13 and an arrow pointing to the right (from -17, -15, -13, -11, -9, -7, -5 with a closed circle at -13 and arrow right)