Question

33.) solve the given inequalities. graph the solutions on the number line provided.
a.) $2x - 6 < 12$
b.) $-5x \geq x + 8$
c.) $-4 < x - 1 < 3$
d.) $x - 6 < -6$ or $2x - 5 \geq 7$

Explanation:

Part a)

Step1: Add 6 to both sides

To isolate the term with \(x\), we add 6 to both sides of the inequality \(2x - 6 < 12\). This gives us \(2x - 6 + 6 < 12 + 6\), which simplifies to \(2x < 18\).

Step2: Divide by 2

Now, we divide both sides of the inequality \(2x < 18\) by 2 to solve for \(x\). So, \(\frac{2x}{2} < \frac{18}{2}\), which simplifies to \(x < 9\).

Step1: Subtract \(x\) from both sides

Starting with the inequality \(-5x \geq x + 8\), we subtract \(x\) from both sides to get \(-5x - x \geq x + 8 - x\). This simplifies to \(-6x \geq 8\).

Step2: Divide by -6 (and reverse inequality)

When we divide both sides of the inequality \(-6x \geq 8\) by -6, we must reverse the inequality sign. So, \(\frac{-6x}{-6} \leq \frac{8}{-6}\), which simplifies to \(x \leq -\frac{4}{3}\) (or \(x \leq -1.\overline{3}\)).

Step1: Add 1 to all parts

For the compound inequality \(-4 < x - 1 < 3\), we add 1 to all three parts. This gives us \(-4 + 1 < x - 1 + 1 < 3 + 1\), which simplifies to \(-3 < x < 4\).

Answer:

The solution to \(2x - 6 < 12\) is \(x < 9\). To graph this, we draw an open circle at 9 on the number line and shade the region to the left of 9.

Part b)