Question

inequalities
symbol words open or closed

greater than open

< less than open
≥ greater than or equal closed
≤ less than or equal closed
remember: if you multiply or divide by a negative number, then switch the inequality
examples:
solve and graph the inequality

1. - 3x + 11 ≥ 44
2. $\frac{x}{2}-5geq - 8$

compound inequalities
definition graphing
and both inequalities are true
or either inequality is true

1. 3x < - 12 or x + 29 ≥ 32
2. x - 8 > - 7 and 5x < 20

Explanation:

Step1: Solve -3x + 11 ≥ 44

Subtract 11 from both sides: -3x + 11-11 ≥ 44 - 11, which simplifies to -3x ≥ 33.

Step2: Isolate x

Divide both sides by -3 and reverse the inequality sign: x ≤ - 11.

Step3: Solve $\frac{x}{2}-5\geq - 8$

Add 5 to both sides: $\frac{x}{2}-5 + 5\geq - 8+5$, which simplifies to $\frac{x}{2}\geq - 3$.

Step4: Isolate x

Multiply both sides by 2: x ≥ - 6.

Step5: Solve 3x < - 12 or x + 29 ≥ 32

For 3x < - 12, divide both sides by 3: x < - 4. For x + 29 ≥ 32, subtract 29 from both sides: x ≥ 3.

Step6: Solve x - 8 > - 7 and 5x < 20

For x - 8 > - 7, add 8 to both sides: x > 1. For 5x < 20, divide both sides by 5: x < 4. So 1 < x < 4.

Answer:

1. x ≤ - 11; 6) x ≥ - 6; 7) x < - 4 or x ≥ 3; 8) 1 < x < 4