Question

1. $-8|7 - 7x| < -56$

Explanation:

Step1: Simplify the inequality

Divide both sides of the inequality \(-8|7 - 7x| < -56\) by \(-8\). Remember that when dividing an inequality by a negative number, the direction of the inequality sign changes. So we get \(|7 - 7x|>7\).

Step2: Split into two inequalities

For \(|a|>b\) (\(b > 0\)), we have \(a>b\) or \(a < -b\). Here \(a = 7-7x\) and \(b = 7\).

• First case: \(7-7x>7\)

Subtract \(7\) from both sides: \(-7x>0\). Then divide both sides by \(-7\) (and reverse the inequality sign), we get \(x < 0\).

• Second case: \(7-7x < -7\)

Subtract \(7\) from both sides: \(-7x<-14\). Divide both sides by \(-7\) (and reverse the inequality sign), we get \(x>2\).

Answer:

The solution of the inequality is \(x < 0\) or \(x>2\). In interval notation, the solution is \((-\infty,0)\cup(2,\infty)\). To graph it, we draw an open circle at \(x = 0\) and shade to the left, and draw an open circle at \(x = 2\) and shade to the right on the number - line.