Question

select the correct answer. which graph represents the solution to this inequality? (-\frac{1}{4}(12x + 8) leq -2x + 11) a. graph a b. graph b c. graph c d. graph d

Explanation:

Step1: Simplify the left side

First, distribute \(-\frac{1}{4}\) to \(12x\) and \(8\). So, \(-\frac{1}{4}(12x + 8)=-3x - 2\). The inequality becomes \(-3x - 2\leq - 2x+11\).

Step2: Solve for \(x\)

Add \(3x\) to both sides: \(-3x - 2+3x\leq - 2x + 11+3x\), which simplifies to \(-2\leq x + 11\). Then subtract \(11\) from both sides: \(-2-11\leq x+11 - 11\), so \(-13\leq x\) (or \(x\geq - 13\)).

Step3: Analyze the graphs

We need a number line with a closed circle at \(-13\) (since the inequality is \(\leq\), or in our solved form \(x\geq - 13\), the circle is closed) and the arrow pointing to the right (since \(x\) is greater than or equal to \(-13\)). Looking at the options, option B has a closed circle at \(-13\) and the arrow pointing right.

Answer:

B