Question: Solve each of the given compound inequalities. Enter your answers using interval notation. \[ 7x…

Solve each of the given compound inequalities. Enter your answers using interval notation.

\[ 7x + 3 > 17 \quad \text{or} \quad 3x - 7 > 14 \]

Solution:

\[ 7x + 3 > 17 \quad \text{and} \quad 3x - 7 > 14 \]

Solution:

Solution

Solve each of the given compound inequalities and enter the answers using interval notation. First, solve the disjunction: 1. \(7x + 3 > 17\) Subtract 3 from both sides: \[ 7x > 14 \] Divide by 7: \[ x > 2 \] So, the solution for \(7x + 3 > 17\) is \(x > 2\). 2. \(3x - 7 > 14\) Add 7 to both sides: \[ 3x > 21 \] Divide by 3: \[ x > 7 \] So, the solution for \(3x - 7 > 14\) is \(x > 7\). This is a disjunction, so combine the solutions: \[ x > 2 \quad \text{or} \quad x > 7 \] The solution in interval notation is \((- \infty, \infty)\). Next, solve the conjunction: 1. \(7x + 3 > 17\) Subtract 3 from both sides: \[ 7x > 14 \] Divide by 7: \[ x > 2 \] 2. \(3x - 7 > 14\) Add 7 to both sides: \[ 3x > 21 \] Divide by 3: \[ x > 7 \] This is a conjunction, so take the intersection of solutions: \[ x > 2 \quad \text{and} \quad x > 7 \] The solution in interval notation is \((7, \infty)\).