Question

solving and graphing a compound inequality select the graph that represents the solution of the compound inequality $-6 \leq 4x + 6 < 14$

Explanation:

Step1: Subtract 6 from all parts

To solve the compound inequality \(-6 \leq 4x + 6 < 14\), we first subtract 6 from each part. This gives us \(-6 - 6 \leq 4x + 6 - 6 < 14 - 6\), which simplifies to \(-12 \leq 4x < 8\).

Step2: Divide all parts by 4

Next, we divide each part of the inequality by 4 to solve for \(x\). So, \(\frac{-12}{4} \leq \frac{4x}{4} < \frac{8}{4}\), which simplifies to \(-3 \leq x < 2\).

Now, we analyze the number line graphs:

• The solution \( -3 \leq x < 2\) means we have a closed circle at \(-3\) (because of the "less than or equal to" sign) and an open circle at \(2\) (because of the "less than" sign), with the line segment connecting them.

Looking at the options, the fourth graph (the last one) has a closed circle at \(-3\) and an open circle at \(2\), with the line between them, which matches our solution.

Answer:

The graph with a closed circle at \(-3\), an open circle at \(2\), and the line segment connecting \(-3\) (inclusive) to \(2\) (exclusive) (the fourth graph option).