Solve the following compound inequality. Write the solution set using interval notation and graph it. \[ \begin{array}{l}1-x<7+x \text { or } 7 x+6>x\end{array} \]

To solve the compound inequality: \[ \begin{array}{l} 1 - x < 7 + x \quad \text{or} \quad 7x + 6 > x \end{array} \] we'll solve each inequality separately and then combine their solutions using the **OR** condition. ### Solving the First Inequality: \(1 - x < 7 + x\) 1. **Subtract 1 from both sides:** \[ -x < 6 + x \] 2. **Subtract \(x\) from both sides:** \[ -2x < 6 \] 3. **Divide both sides by \(-2\) (remember to reverse the inequality sign when dividing by a negative number):** \[ x > -3 \] **Solution for the first inequality:** \(x > -3\) --- ### Solving the Second Inequality: \(7x + 6 > x\) 1. **Subtract \(x\) from both sides:** \[ 6x + 6 > 0 \] 2. **Subtract 6 from both sides:** \[ 6x > -6 \] 3. **Divide both sides by 6:** \[ x > -1 \] **Solution for the second inequality:** \(x > -1\) --- ### Combining the Solutions Since the compound inequality uses **OR**, we take the **union** of the two solution sets: - \(x > -3\) **OR** \(x > -1\) The union of these two conditions is simply \(x > -3\), because any \(x\) that satisfies \(x > -1\) also satisfies \(x > -3\). ### Final Solution - **Interval Notation:** \((-3, \infty)\) - **Graph:**  - An open circle is placed at \(-3\). - A shaded line extends to the right from \(-3\) towards infinity.