Question

solve each inequality. graph your solution on a number line.

1. x + 7 < 12
2. -3y ≥ 9

notes
compound inequalities combine two inequalities with and or or.

• and (intersection):

both conditions must be true at the same time.
example: 2 < x ≤ 7 means x is greater than 2 and less than or equal to 7.
graph:

• or (union):

at least one condition must be true.
example: x < -3 or x ≥ 5.
graph:

Explanation:

Step1: Solve the first inequality

Subtract 7 from both sides of \(x + 7<12\).
\(x+7 - 7<12 - 7\), so \(x < 5\).
On the number - line, we draw an open circle at 5 (since \(x\) is less than, not less than or equal to 5) and shade to the left.

Step2: Solve the second inequality

Divide both sides of \(-3y\geq9\) by - 3. When dividing an inequality by a negative number, the direction of the inequality sign changes.
\(\frac{-3y}{-3}\leq\frac{9}{-3}\), so \(y\leq - 3\).
On the number - line, we draw a closed circle at - 3 (since \(y\) is less than or equal to - 3) and shade to the left.

Answer:

1. Solution of \(x + 7<12\) is \(x < 5\). Graph: Open circle at 5, shade to the left.
2. Solution of \(-3y\geq9\) is \(y\leq - 3\). Graph: Closed circle at - 3, shade to the left.