To solve inequalities, we rely on fundamental properties discussed in Section on Inequalities . Two of the most important are:
- We can add (or subtract) the same quantity from both sides without changing the direction of the inequality.
- We can multiply (or divide) both sides by a positive quantity, without changing the direction of the inequality.
- If we multiply or divide both sides of an inequality by a negative quantity, the direction of the inequality reverses (see Section on Inequalities ).
Caution: Never multiply or divide both sides of an inequality by a quantity whose sign is unknown!
Linear inequalities are often easy to solve. We just need to isolate the variable on one side of the inequality sign.
Example 1. Solve the following inequality
\[7x-5\geq4x+4\]
Solution
\[\begin{align} 7x-5 & \geq4x+4\tag{given inequality}\\ 7x-4x-5 & \ge4x-4x+4\tag{subtract $4x$ from both sides}\\ 3x-5 & \ge4\tag{simplify}\\ 3x & \geq9\tag{add $5$ to both sides}\\ x & \geq3\tag{divide both sides by $3$} \end{align}\]
Example 2. Solve: \[-4<\dfrac{7-5x}{2}\leq1\]
Solution
\[-4<\frac{7-5x}{2}\leq1\] \[-8<7-5x\leq2\tag{multiply by $2$}\] \[-15<-5x\leq-5\tag{subtract $7$}\] \[3>x\geq1\tag{divide by $-5$}\] which can alternatively be rewritten as \(1\leq x<3\) . For the last step, recall that when we divide both sides of an inequality by a negative number, the direction of the inequality changes (see Section on Inequalities ).
- For the last example, note that \[-4<\dfrac{7-5x}{2}\leq1\] means \[-4<\dfrac{7-5x}{2}\qquad\text{and}\qquad\dfrac{7-5x}{2}\leq1.\] In fact, we have to solve two inequalities.