Lesson 2.7: Solve Linear Inequalities

New Concept

Solving linear inequalities means finding all values for a variable that make the inequality true. You'll learn to use algebraic properties to find these solutions and represent them visually on a number line and with interval notation.

What’s next

Up next, you'll walk through interactive examples, and then apply these rules in a series of practice cards and challenge problems.

Property

Any number greater than 3 is a solution to the inequality $x > 3$.
We show the solutions to the inequality $x > 3$ on the number line by shading in all the numbers to the right of 3.
Because the number 3 itself is not a solution, we put an open parenthesis at 3.
The graph of the inequality $x \geq 3$ is very much like the graph of $x > 3$, but now we need to show that 3 is a solution, too.
We do that by putting a bracket at $x = 3$. The open parentheses symbol, (, shows that the endpoint of the inequality is not included.
The open bracket symbol, [, shows that the endpoint is included. We can also represent inequalities using interval notation.
The inequality $x > 3$ is expressed as $(3, \infty)$.
The symbol $\infty$ is read as 'infinity'.
The inequality $x \leq 1$ is written in interval notation as $(-\infty, 1]$.
The symbol $-\infty$ is read as 'negative infinity'.

Examples

• The inequality $x < 4$ includes all numbers to the left of 4. On a number line, we place a parenthesis at 4 and shade to the left. In interval notation, this is $(-\infty, 4)$.

• The inequality $y \geq -2$ includes -2 and all numbers to its right. On a number line, we place a bracket at -2 and shade to the right. In interval notation, this is $[-2, \infty)$.

Property

Subtraction Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a - c < b - c$.
if $a > b$, then $a - c > b - c$.

Addition Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a + c < b + c$.
if $a > b$, then $a + c > b + c$.

Examples

• To solve $x + 7 \leq 15$, subtract 7 from both sides. This gives $x \leq 8$. The solution is all numbers less than or equal to 8, or $(-\infty, 8]$.

Property

For any real numbers $a$, $b$, $c$
if $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
if $a > b$ and $c > 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a < b$ and $c < 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
When we divide or multiply an inequality by a:
• positive number, the inequality stays the same.
• negative number, the inequality reverses.

Examples

• To solve $6x > 48$, divide both sides by 6. Since 6 is positive, the inequality stays the same: $x > 8$.

• To solve $-4y \geq 20$, divide both sides by -4. Since -4 is negative, the inequality reverses: $y \leq -5$.

Property

To solve a linear inequality, simplify each side as much as possible using distribution and combining like terms.
Then, use the Addition and Subtraction Properties of Inequality to collect variable terms on one side and constant terms on the other.
Finally, use the Multiplication or Division Properties of Inequality to isolate the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number.

Examples

• To solve $3x + 5 > 20$, subtract 5 from both sides to get $3x > 15$. Then, divide by 3 to get $x > 5$.

• To solve $7p - 2 \leq 3p + 10$, subtract $3p$ from both sides to get $4p - 2 \leq 10$. Add 2 to both sides to get $4p \leq 12$. Divide by 4 to get $p \leq 3$.

Property

To translate sentences into inequalities, identify keywords that describe the relationship between quantities.
'is no more than' translates to $\leq$.
'is at least' translates to $\geq$.
'exceeds' or 'is greater than' translates to $>$.
'is smaller than' or 'is less than' translates to $<$.
'is at most' translates to $\leq$.

Examples

• 'Eight times a number n is no more than 48' translates to $8n \leq 48$. Dividing by 8 gives $n \leq 6$.

• 'Twenty less than a number x is at least 50' translates to $x - 20 \geq 50$. Adding 20 gives $x \geq 70$.