Question

7(8 - 8x) ≥ -7(-8 + 6x) graph the solution: number line options

Explanation:

Step1: Expand both sides

First, we expand the left - hand side and the right - hand side of the inequality \(7(8 - 8x)\geq- 7(-8 + 6x)\).
Using the distributive property \(a(b + c)=ab+ac\), the left - hand side \(7(8 - 8x)=7\times8-7\times8x = 56-56x\).
The right - hand side \(-7(-8 + 6x)=(-7)\times(-8)+(-7)\times6x=56 - 42x\).
So the inequality becomes \(56-56x\geq56 - 42x\).

Step2: Subtract 56 from both sides

Subtract 56 from both sides of the inequality \(56-56x\geq56 - 42x\).
\(56-56x - 56\geq56 - 42x-56\), which simplifies to \(-56x\geq - 42x\).

Step3: Add 56x to both sides

Add \(56x\) to both sides of the inequality \(-56x\geq - 42x\).
\(-56x + 56x\geq - 42x+56x\), which gives \(0\geq14x\).

Step4: Divide both sides by 14

Divide both sides of the inequality \(0\geq14x\) by 14. Since 14 is a positive number, the direction of the inequality sign remains the same.
\(\frac{0}{14}\geq\frac{14x}{14}\), so \(0\geq x\) or \(x\leq0\).

Answer:

The solution to the inequality \(7(8 - 8x)\geq-7(-8 + 6x)\) is \(x\leq0\). To graph this solution on a number line, we draw a closed circle at \(x = 0\) (because the inequality is non - strict, \(\leq\)) and shade the region to the left of \(0\) (representing all values of \(x\) that are less than or equal to \(0\)).