Solve the following compound inequality. Write the solution set using interval notation and graph it. \[ 1-x<7+x \text { or } 6 x+5>x \]

To solve the compound inequality \( 1 - x < 7 + x \) or \( 6x + 5 > x \), we will tackle each part separately. **First inequality:** Starting with \( 1 - x < 7 + x \): 1. Add \( x \) to both sides: \[ 1 < 7 + 2x \] 2. Subtract \( 7 \) from both sides: \[ -6 < 2x \] 3. Divide by \( 2 \): \[ -3 < x \] or \( x > -3 \). **Second inequality:** Now let's solve \( 6x + 5 > x \): 1. Subtract \( x \) from both sides: \[ 6x - x + 5 > 0 \] 2. This simplifies to: \[ 5x + 5 > 0 \] 3. Subtract \( 5 \): \[ 5x > -5 \] 4. Divide by \( 5 \): \[ x > -1 \]. **Combining the results:** The overall solution for the compound inequality is either \( x > -3 \) or \( x > -1 \). Since \( x > -1 \) is the more restrictive condition, our solution can be written as: \[ x > -1 \]. **Interval notation:** The solution set in interval notation is: \[ (-1, \infty) \]. **Graph:** To graph this inequality, you would make an open circle at \(-1\) on the number line indicating that \(-1\) is not included, and shade all the values to the right of \(-1\) towards infinity.