Question

solve for x and graph the solution on the number line below. if possible, resolve your answer to a single inequality. in case of no solution (∅), leave the number line blank.

4x + 10 ≤ 22 or 4x + 10 < 46

answer attempt 1 out of 5

inequality notation:

number line:
number line image with -12 to 12, click and drag to plot line

Explanation:

Step1: Solve \(4x + 10 \leq 22\)

Subtract 10 from both sides: \(4x + 10 - 10 \leq 22 - 10\)
Simplify: \(4x \leq 12\)
Divide both sides by 4: \(\frac{4x}{4} \leq \frac{12}{4}\)
Result: \(x \leq 3\)

Step2: Solve \(4x + 10 < 46\)

Subtract 10 from both sides: \(4x + 10 - 10 < 46 - 10\)
Simplify: \(4x < 36\)
Divide both sides by 4: \(\frac{4x}{4} < \frac{36}{4}\)
Result: \(x < 9\)

Step3: Analyze the "or" condition

The solution to \(4x + 10 \leq 22\) or \(4x + 10 < 46\) is the union of the two solution sets. Since all \(x \leq 3\) are also \(x < 9\) (because 3 < 9), the combined solution is \(x < 9\) (because the "or" takes the broader set; the set \(x \leq 3\) is a subset of \(x < 9\), so the union is \(x < 9\)).

Answer:

Inequality Notation: \(x < 9\)

(For the number line: Draw an open circle at 9 (since it's \(x < 9\), not \(x \leq 9\)) and shade the line to the left of 9.)