Question

1. - 4 + 9|4x - 7| ≥ 113
2. 8 - 4|7 - 7x| > - 76

Explanation:

Step1: Isolate the absolute - value term for the first inequality

For the inequality \(-4 + 9|4x - 7|\geq113\), first add 4 to both sides:
\(9|4x - 7|\geq113 + 4\), so \(9|4x - 7|\geq117\). Then divide both sides by 9: \(|4x - 7|\geq\frac{117}{9}=13\).
This gives two cases:
Case 1: \(4x-7\geq13\). Add 7 to both sides: \(4x\geq13 + 7=20\), then divide by 4, \(x\geq5\).
Case 2: \(4x - 7\leq - 13\). Add 7 to both sides: \(4x\leq-13 + 7=-6\), then divide by 4, \(x\leq-\frac{3}{2}\).

Step2: Isolate the absolute - value term for the second inequality

For the inequality \(8-4|7 - 7x|>-76\), first subtract 8 from both sides: \(-4|7 - 7x|>-76 - 8=-84\).
Divide both sides by - 4. When dividing an inequality by a negative number, the direction of the inequality sign changes. So \(|7 - 7x|<21\).
This gives two cases:
Case 1: \(7-7x<21\). Subtract 7 from both sides: \(-7x<21 - 7 = 14\), then divide by - 7 and reverse the inequality sign, \(x>-2\).
Case 2: \(7-7x>-21\). Subtract 7 from both sides: \(-7x>-21 - 7=-28\), then divide by - 7 and reverse the inequality sign, \(x < 4\).

Answer:

For \(-4 + 9|4x - 7|\geq113\), \(x\leq-\frac{3}{2}\) or \(x\geq5\).
For \(8-4|7 - 7x|>-76\), \(-2