GMAT inequalities strategies

Treat inequalities like equations, with one exception: when you multiply or divide both sides by a negative number, flip the inequality sign. If $-2x > 6$, dividing by $-2$ gives $x < -3$, not $x > -3$. You can add and subtract freely without flipping. You can multiply or divide by a positive without flipping. Only a negative multiplier or divisor flips the sign.

An absolute value inequality splits into two cases. If $|x| < 5$, then $-5 < x < 5$. The 'less than' stays a compound inequality. If $|x| > 5$, then $x < -5$ or $x > 5$. The 'greater than' splits into two regions. The rule: less-than becomes a between, greater-than becomes an outside. For a more complex form like $|2x - 3| \leq 7$, write $-7 \leq 2x - 3 \leq 7$, then solve the compound inequality in one pass: add 3 to get $-4 \leq 2x \leq 10$, divide by 2 to get $-2 \leq x \leq 5$.

Solve for $x$: $|3x - 6| > 9$. Step 1: Split into two cases. Because the inequality is 'greater than,' the solution is two regions: $3x - 6 > 9$ or $3x - 6 < -9$. Step 2: Solve the first case. $3x - 6 > 9$ becomes $3x > 15$, then $x > 5$. Step 3: Solve the second case. $3x - 6 < -9$ becomes $3x < -3$, then $x < -1$. Step 4: Combine. The solution is $x < -1$ or $x > 5$. Test a value: $x = 0$ gives $|0 - 6| = 6$, which is not greater than 9, and 0 is between $-1$ and 5, so it is correctly excluded. The answer is $x < -1$ or $x > 5$.