Question

use the number lines provided to graph the solution set for the inequalities.
things to keep in mind:

• did you solve for the variable?
• should the sign flip or stay the same?
• will you use a circle or a dot on the value?
• did you check your number line by testing out a value?

3x ≥ x - 8
\frac{x + 5}{2} < 4

Explanation:

Step1: Solve the first inequality \(3x\geq x - 8\)

Subtract \(x\) from both sides: \(3x-x\geq x - 8-x\), which simplifies to \(2x\geq - 8\).
Then divide both sides by \(2\): \(x\geq\frac{-8}{2}=-4\).
On the number - line, we use a closed dot (since the inequality is \(\geq\)) at \(-4\) and shade to the right.

Step2: Solve the second inequality \(\frac{x + 5}{2}<4\)

Multiply both sides by \(2\): \(x + 5<8\).
Subtract \(5\) from both sides: \(x<8 - 5=3\).
On the number - line, we use an open circle (since the inequality is \(<\)) at \(3\) and shade to the left.

Answer:

For \(3x\geq x - 8\), the solution is \(x\geq - 4\). Graph: closed dot at \(-4\) and shade to the right.
For \(\frac{x + 5}{2}<4\), the solution is \(x<3\). Graph: open circle at \(3\) and shade to the left.