Question

solve the inequality. express your answer using set notation and interval notation. graph the solution set. -4(x + 3) > 16 the solution is expressed in set notation as {x | \\(\square\\)} the solution is expressed in interval notation as \\(\square\\). choose the correct graph of the solution set below. a. number line graph b. number line graph c. number line graph d. number line graph

Explanation:

Step1: Solve the inequality

Start with \(-4(x + 3) > 16\). Divide both sides by \(-4\). Remember that when dividing an inequality by a negative number, the inequality sign flips. So we get \(x + 3 < \frac{16}{-4}\), which simplifies to \(x + 3 < -4\).

Step2: Isolate x

Subtract 3 from both sides: \(x < -4 - 3\), so \(x < -7\).

Step3: Set notation

The solution in set notation is \(\{x | x < -7\}\).

Step4: Interval notation

In interval notation, this is \((-\infty, -7)\).

Step5: Graph identification

For the graph, we need an open circle (since \(x < -7\), not \(x \leq -7\)) at \(-7\) and the arrow pointing to the left (because \(x\) is less than \(-7\)). Looking at the options, option D has an open circle (or the correct endpoint style) and the arrow to the left for \(x < -7\). Wait, let's re - check:

• Option A: Arrow to the right from \(-8\)? No.
• Option B: Arrow to the right from \(-8\)? No.
• Option C: Arrow to the left but the endpoint? Wait, no. Wait, our solution is \(x < -7\). So the graph should have a parenthesis (open circle) at \(-7\) and the line going to the left. Looking at the options, option D: Let's see the number line. The correct graph for \(x < -7\) is a line with an open circle at \(-7\) and the arrow pointing to the left (towards more negative numbers). So the correct graph is D.

Answer:

Set notation: \(\{x | x < -7\}\)
Interval notation: \((-\infty, -7)\)
Graph: Option D (the graph with the arrow to the left and the endpoint at \(-7\) with an open circle - like representation as per the option D's graph)