Two-Step Inequalities Worksheets

Sometimes we are confronted with a problem that contains words such as less than, more than, at least and at most. These problems when translated mathematically will lead to inequalities. These inequalities may either require one step, two steps or even multi steps. Inequalities requiring two steps in solving them are called two-step inequalities. Inequalities such as $$2x+3 < 5$$ and $$3x-5\geq 7$$ are examples of two-step inequalities.

Just like solving two–step equations, there is no definite rule “which” operation to undo first in solving two-step inequalities. Nonetheless, listed below are some helpful tips on how to solve two-step inequalities.

1. Isolate the variable term on one side and the constant term on the other by using addition property.
2. Isolate the variable and its numerical coefficient by using multiplication property.
3. Write the solution set in either of the forms: interval, set notation or graph.
EXAMPLES

Solve the following inequalities. Write the solution set in both interval and graphical forms.

\begin{align*} 1.\;\;\; \displaystyle &3x-4 >5\\&3x-4+4>5+4 &&\text{add 4 to both sides} \\&3x>9 &&\text{multiply both sides by 1/3}\\ &\therefore x>3\\ \\&\textbf{Solution set:}\; \left ( 3,+\infty \right ) \end{align*}

\begin{align*} 2.\;\;\; \displaystyle &3x+4 \leq 1 \\&3x+4-4\leq 1-4 &&\text{add -4 to both sides} \\&3x\leq -3 &&\text{multiply both sides by 1/3}\\ &\therefore x\leq -1\\ \\&\textbf{Solution set:}\; \left ( -\infty,-1 \right ] \end{align*}

\begin{align*} 3.\;\;\; \displaystyle &2x++ < -2 \\&-2x+6-6 < -2-6 &&\text{add -6 to both sides} \\&-2x < -8 &&\text{multiply both sides by -1/2}\\ &\therefore x > 4\\ \\ &\textbf{Solution set:}\; \left ( 4,+\infty \right ) \end{align*}

\begin{align*} 4.\;\;\; \displaystyle &-5x \leq 6 \\&1-5x \leq 6-1 &&\text{add -1 to both sides} \\&-5x < 5 &&\text{multiply both sides by -1/5}\\ &\therefore x \geq -1\\ \\ &\textbf{Solution set:}\; \left [ -1,+\infty \right ) \end{align*}