The Simplest Way to Learn Multistep Inequalities

Simplify Terms on Separate Sides

1). Apply the distributive property to eradicate parentheses, if any exist. For instance, consider the inequality 6 (m -- 12) > 15m -- 3m + 24. Multiply the 6 by both the m and the -12 to obtain 6m -- 72 > 15m -- 3m + 24.
2). Combine any like terms existing on the same side of the inequality. In the example, on the right-hand side, 15m and -3m are like terms; subtract them to simplify the inequality to 6m -- 72 > 12m + 24.
3). Check your work so far to ensure correctness because otherwise, the remainder of the problem will be solved improperly.

1). Transfer any constant terms on the left side of the inequality to its right side, by subtracting them from or adding them to constant terms already existing on the right. As when solving regular equations, add negative terms and subtract positive ones. In 6m -- 72 > 12m + 24, the constant on the left, -72, is negative, so add it to both sides to get 6m > 12m + 96.
2). Transfer any variable terms on the right side of the inequality to its left side, by subtracting them from or adding them to variable terms already appearing on the left. As in the previous step, add negative terms and subtract positive ones. In 6m > 12m + 96, subtract 12m from both sides, obtaining -6m > 96.
3). Multiply or divide both sides by the coefficient. Just as when you're solving equations, if the current operation between the variable and its coefficient is division, then multiply, but if the current operation is multiplication, then divide. In -6m > 96, -6m is a multiplicative operation; thus, divide both sides by -6. In this case, a particular rule for operations on inequalities must be used. This rule requires that when dividing or multiplying by a negative coefficient, the direction of the inequality symbol must be reversed. In the example, change the greater-than sign into a less-than sign while dividing by the -6. So the solution to the example is m < -16.