Example Question - inequality with 'or'

Solving a Compound Inequality with 'Or'

The image shows a compound inequality that consists of two parts connected by "or". The first part of the inequality is: \[ 6 + 7m \leq 6m - 5 \] The second part of the inequality is: \[ 3m - 7 < 5 + 6m \] To solve the first part of the inequality, we will isolate the variable \( m \) on one side: \[ 6 + 7m \leq 6m - 5 \] Subtract \( 6m \) from both sides: \[ 7m - 6m \leq -5 - 6 \] Combine like terms: \[ m \leq -11 \] Now to solve the second part of the inequality: \[ 3m - 7 < 5 + 6m \] Subtract \( 3m \) from both sides: \[ -7 < 5 + 3m \] Subtract \( 5 \) from both sides to isolate the variable term: \[ -12 < 3m \] Divide both sides by \( 3 \) to solve for \( m \): \[ -4 < m \] Now we have two parts to the solution, and since they are connected by "or," we will take the union of the two solutions. The solutions to the compound inequality are: \[ m \leq -11 \quad \text{or} \quad m > -4 \] This means that any \( m \) that is less than or equal to -11 or greater than -4 satisfies the original compound inequality.