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College Algebra/Solving Linear Equations and Inequalities/Solving Multistep Inequalities

Solving Linear Equations and Inequalities

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Solving Multistep Inequalities

Properties of inequality and properties of operations can be combined to solve inequalities using multiple steps.
As with equations, some inequalities require two or more steps to find the solution. Start by simplifying each side, if necessary. Apply the addition or subtraction property of inequality to isolate the term with the variable. Then apply the multiplication or division property to isolate the variable.
Step-By-Step Example
Solving Multistep Inequalities
Solve the inequality:
4x−2<7+x4x-2 \lt 7+x
Step 1
Apply the subtraction property of inequality to combine variable terms on one side of the inequality. Subtract xx from both sides.
4x−2<7+x4x−2−x<7+x−x3x−2<7\begin{aligned} 4x-2&\lt 7+x\\4x-2-x&\lt 7+x-x\\3x-2&\lt 7\end{aligned}
Step 2
Apply the addition property of inequality to undo the subtraction. Add 2 to both sides.
3x−2<73x−2+2<7+23x<9\begin{aligned}3x-2&\lt 7\\3x-2+2&\lt 7+2\\3x&\lt 9\end{aligned}
Solution
Apply the division property of inequality to undo the multiplication. Divide both sides by 3.
3x<93x3<93x<3\begin{aligned}3x&\lt 9\\\frac{3x}{3}&\lt \frac{9}{3}\\x&\lt 3\end{aligned}
<Properties of Inequality>Absolute Value Equations