Question

1. use the inequality 18 < -3(4x - 2).

a. solve the inequality for x.
b. which graph shows the solution to the inequality?

Explanation:

Part a: Solve the inequality \( 18 < -3(4x - 2) \) for \( x \)

Step 1: Distribute the -3

We start with the inequality \( 18 < -3(4x - 2) \). First, we distribute the -3 on the right side. Using the distributive property \( a(b - c)=ab - ac \), we get:
\( 18 < -12x + 6 \)

Step 2: Subtract 6 from both sides

To isolate the term with \( x \), we subtract 6 from both sides of the inequality. This gives:
\( 18 - 6 < -12x + 6 - 6 \)
Simplifying both sides, we have:
\( 12 < -12x \)

Step 3: Divide by -12 (and reverse inequality)

Now, we divide both sides by -12. When we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. So:
\( \frac{12}{-12} > \frac{-12x}{-12} \)
Simplifying both sides, we get:
\( - 1>x \)
Or, equivalently, \( x < - 1 \)

Part b: Determine the correct graph

The solution to the inequality \( x < - 1 \) means we are looking for all numbers less than -1. On a number line, this is represented by an open circle at -1 (since -1 is not included in the solution, because the inequality is strict, \( x < - 1 \), not \( x\leq - 1 \)) and an arrow pointing to the left (indicating all numbers less than -1).

Looking at the graphs:

• Graph A: Open circle at a number, arrow to the right (represents \( x > \text{[number]} \))
• Graph B: Open circle at -1, arrow to the left (represents \( x < - 1 \))
• Graph C: Closed circle at a number, arrow to the left (represents \( x\leq \text{[number]} \))
• Graph D: Closed circle at a number, arrow to the right (represents \( x\geq \text{[number]} \))

So the graph that shows the solution \( x < - 1 \) is Graph B.

Answer:

s:
a. \( x < -1 \)
b. B (the graph with an open circle at -1 and an arrow pointing to the left)