Linear Inequalities

Linear inequalities are often easy to solve. We just need to isolate the variable on one side of the inequality sign.

• Recall that we can
• add (or subtract) the same quantity (+ or $-$) from both sides of the inequality, or
• multiply (or divide) both sides of the inequality by a positive quantity, and the inequality will still hold.
• If we multiply or divide both sides of an inequality by a negative quantity, the direction of the inequality will be reversed (See).

Never multiply or divide both sides of an inequality by a quantity whose sign is unknown!

Example 1
Solve each inequality
(a) $7x-5\geq4x+4$
(b) $-4<\dfrac{7-5x}{2}\leq1$
Solution
(a)
\begin{align*}
\begin{array}{rlr}
7x-5 & \geq4x+4 & {\small\text{ (given inequality)}}\\
7x-4x-5 & \ge4x-4x+4 &{\small\text{ (subtract $4x$ from both sides)}}\\
3x-5 & \ge4 &{\small \text{(simplify)}}\\
3x & \geq9 &{\small\text{ (add 5 to both sides)}}\\
x & \geq3 &{\small \text{(divide both sides by 3)}}
\end{array}
\end{align*}
(b)
$-4<\frac{7-5x}{2}\leq1$

Multiply each term by 2:
$-8<7-5x\leq2$

Subtract 7 from each term:
$-15<-5x\leq-5$ Divide each term by $-5$:
$3>x\geq 1$ 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 1.4).

The graph of $y=-\frac{5}{2}x+\frac{7}{2}$ shows that for $1\leq x<3$, we have $-4<y\le1$.