Question

solve each inequality below. darken in the solution (the region satisfying all inequalities). show your work.

1. ( 0 geq x^2 - 4x - 5 )
2. ( |x + 1| geq 3 )
3. ( 3x - 6 < -x + 2 )

Explanation:

Step1: Solve the inequality \(3x - 6 < -x + 2\)

First, we want to get all the \(x\) terms on one side and the constants on the other side. Add \(x\) to both sides of the inequality: \(3x + x- 6 < -x+x + 2\), which simplifies to \(4x-6 < 2\). Then, add 6 to both sides: \(4x-6 + 6<2 + 6\), so \(4x<8\).

Step2: Solve for \(x\)

Divide both sides of the inequality \(4x < 8\) by 4. Since 4 is a positive number, the direction of the inequality sign remains the same. So \(\frac{4x}{4}<\frac{8}{4}\), which gives \(x < 2\).

Answer:

The solution to the inequality \(3x - 6 < -x + 2\) is \(x < 2\)