Questions: Solve the compound inequality and give your answer in interval notation. [ 7x+1>2x+6 text OR 3(-6x-6)+9 geq -15x+6 ]

Solution Steps

To solve the compound inequality, we need to break it down into two separate inequalities and solve each one individually. Then, we combine the solutions using the union of the intervals.

1. Solve the first inequality: \(7x + 1 > 2x + 6\)
2. Solve the second inequality: \(3(-6x - 6) + 9 \geq -15x + 6\)
3. Combine the solutions from both inequalities using the union of the intervals.

The first inequality is: \[ 7x + 1 > 2x + 6 \]

Subtract \(2x\) from both sides: \[ 7x - 2x + 1 > 6 \] \[ 5x + 1 > 6 \]

Subtract 1 from both sides: \[ 5x > 5 \]

Divide both sides by 5: \[ x > 1 \]

The second inequality is: \[ 3(-6x - 6) + 9 \geq -15x + 6 \]

Distribute the 3: \[ -18x - 18 + 9 \geq -15x + 6 \] \[ -18x - 9 \geq -15x + 6 \]

Add \(18x\) to both sides: \[ -9 \geq 3x + 6 \]

Subtract 6 from both sides: \[ -15 \geq 3x \]

Divide both sides by 3: \[ -5 \geq x \] or equivalently, \[ x \leq -5 \]

The compound inequality is: \[ x > 1 \text{ OR } x \leq -5 \]

In interval notation, this is: \[ (-\infty, -5] \cup (1, \infty) \]

Final Answer