Question

1. solve the inequality. represent the solution on the number line. -2x + 4 > -5(4 + 2x)

Explanation:

Step1: Expand the right side

First, we expand \(-5(4 + 2x)\) using the distributive property \(a(b + c)=ab+ac\). So, \(-5(4 + 2x)=-5\times4+(-5)\times2x=-20 - 10x\). The inequality becomes \(-2x + 4>-20 - 10x\).

Step2: Add \(10x\) to both sides

To get all the \(x\) terms on one side, we add \(10x\) to both sides of the inequality. \(-2x+10x + 4>-20 - 10x+10x\), which simplifies to \(8x + 4>-20\).

Step3: Subtract 4 from both sides

Next, we subtract 4 from both sides to isolate the term with \(x\). \(8x+4 - 4>-20 - 4\), so \(8x>-24\).

Step4: Divide by 8

Now, we divide both sides by 8 to solve for \(x\). \(\frac{8x}{8}>\frac{-24}{8}\), which gives \(x>-3\).

Answer:

The solution to the inequality \(-2x + 4>-5(4 + 2x)\) is \(x > - 3\). To represent this on the number line, we draw an open circle at \(-3\) (since \(x\) is not equal to \(-3\)) and shade the region to the right of \(-3\) to indicate all values of \(x\) greater than \(-3\).