Question

1. identify the graph that shows the solutions to the inequality $8(1 + 2x) < 32$.

Explanation:

Step1: Solve the inequality

First, we solve the inequality \(8(1 + 2x) < 32\). Divide both sides by 8:
\(\frac{8(1 + 2x)}{8} < \frac{32}{8}\)
Simplify to get:
\(1 + 2x < 4\)
Subtract 1 from both sides:
\(2x < 4 - 1\)
\(2x < 3\)
Divide both sides by 2:
\(x < \frac{3}{2}\) or \(x < 1.5\)

Step2: Analyze the number lines

Now we analyze the number lines:

• The first two number lines have markings starting at -1, 0, 1, etc., and the solution we found is \(x < 1.5\), so these are not relevant as their scales don't match the solution range (since \(1.5\) is between 1 and 2, but the first two have higher numbers like 6,7,8 which are not related to \(x < 1.5\)).
• The third number line has a filled dot at 1.5 (which would be for \(x \leq 1.5\)), but our solution is \(x < 1.5\) (open dot).
• The fourth number line has an open dot at 1.5 (since \(1.5\) is between 1 and 2, and the open dot indicates \(x < 1.5\)) and the arrow pointing to the left (indicating all numbers less than 1.5), which matches our solution.

Answer:

The fourth number line (the one with -4, -3, -2, -1, 0, 1, 2, 3, 4 markings, open dot at 1.5, and arrow to the left).