Question

3 use distance on a number line to explain why the solutions to |3 - x| ≥ 2 include all numbers less than or equal to 1 as well as all numbers greater than or equal to 5.

4 a. explain what the solutions to the inequality |x - 4.26| ≥ 0 mean based on distance on a number line. what are the solutions?

b. explain what the solutions to the inequality |x - 2| < 0 mean based on distance on a number line. what are the solutions?

5 solve the inequalities. graph the solutions on the number lines.

a. |x - 4| + 2 ≥ 5

number line

b. 2|x + 3| < 10

number line

Explanation:

5a. Solve \(|x - 4|+2\geq5\)

Step1: Simplify the inequality

Subtract \(2\) from both sides of \(|x - 4|+2\geq5\).
\(|x - 4|+2-2\geq5 - 2\), so \(|x - 4|\geq3\).

Step2: Use the definition of absolute - value inequality

If \(|a|\geq b\) (\(b\geq0\)), then \(a\geq b\) or \(a\leq - b\).
Here \(a=x - 4\) and \(b = 3\). So \(x-4\geq3\) or \(x - 4\leq-3\).

Step3: Solve \(x-4\geq3\)

Add \(4\) to both sides of \(x-4\geq3\). \(x-4 + 4\geq3+4\), so \(x\geq7\).

Step4: Solve \(x - 4\leq-3\)

Add \(4\) to both sides of \(x - 4\leq-3\). \(x-4 + 4\leq-3 + 4\), so \(x\leq1\).

Graph:

On the number - line, draw closed circles at \(x = 1\) and \(x = 7\) (because the inequality includes equality) and shade the regions to the left of \(x = 1\) and to the right of \(x = 7\).

5b. Solve \(2|x + 3|\lt10\)

Step1: Simplify the inequality

Divide both sides of \(2|x + 3|\lt10\) by \(2\). \(\frac{2|x + 3|}{2}\lt\frac{10}{2}\), so \(|x + 3|\lt5\).

Step2: Use the definition of absolute - value inequality

If \(|a|\lt b\) (\(b\gt0\)), then \(-b\lt a\lt b\).
Here \(a=x + 3\) and \(b = 5\). So \(-5\lt x+3\lt5\).

Step3: Solve \(-5\lt x + 3\)

Subtract \(3\) from both sides: \(-5-3\lt x+3-3\), so \(-8\lt x\).

Step4: Solve \(x + 3\lt5\)

Subtract \(3\) from both sides: \(x+3-3\lt5-3\), so \(x\lt2\).

Graph:

On the number - line, draw open circles at \(x=-8\) and \(x = 2\) (because the inequality is strict) and shade the region between \(x=-8\) and \(x = 2\).

Answer:

a. The solution of \(|x - 4|+2\geq5\) is \(x\leq1\) or \(x\geq7\).
b. The solution of \(2|x + 3|\lt10\) is \(-8\lt x\lt2\).