Lesson 2.5: Solve Linear Inequalities

New Concept

A linear inequality is a statement comparing linear expressions, like $ax + b \leq c$. In this lesson, we'll master solving these inequalities, graphing their solutions on a number line, translating words into inequalities, and solving real-world applications.

What’s next

First, you'll practice graphing inequalities on the number line. Then, you'll solve them step-by-step through a series of interactive examples and challenge problems.

Property

We show all the solutions to an inequality on the number line by shading. An open parenthesis shows an endpoint is not a solution; a bracket shows an endpoint is a solution. In interval notation, we use parentheses or brackets to show the interval of solutions. The symbol $\infty$ is read as “infinity,” and $-\infty$ is read as “negative infinity.”

Examples

• The inequality $x > 5$ is written in interval notation as $(5, \infty)$. On a number line, an open parenthesis is placed at 5, and the line is shaded to the right.
• The inequality $y \leq -1$ is written in interval notation as $(-\infty, -1]$. On a number line, a bracket is placed at -1, and the line is shaded to the left.
• The inequality $-4 \leq z < 2$ is written as $[-4, 2)$. On a number line, there is a bracket at -4, a parenthesis at 2, and the line between them is shaded.

Explanation

Think of graphing an inequality as drawing a map of all its solutions. A parenthesis '(' or ')' means the endpoint is not included, while a bracket '[' or ']' means it is. Interval notation is just a shorthand for this map.

Property

For any numbers $a$, $b$, and $c$, if $a < b$, then

For any numbers $a$, $b$, and $c$, if $a > b$, then

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

Examples

• To solve $x + 7 < 15$, subtract 7 from both sides: $x + 7 - 7 < 15 - 7$, which simplifies to $x < 8$.
• To solve $y - 4 \geq -2$, add 4 to both sides: $y - 4 + 4 \geq -2 + 4$, which simplifies to $y \geq 2$.
• Given $12 > z + 5$, subtract 5 from both sides: $12 - 5 > z + 5 - 5$, so $7 > z$, which means $z < 7$.

Explanation

This property is just like it is for equations. You can add or subtract the same number on both sides of an inequality, and the relationship between the two sides stays the same. The inequality sign does not change.

Property

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

Examples

• To solve $3x > 21$, divide by 3 (a positive number): $\frac{3x}{3} > \frac{21}{3}$, so $x > 7$. The sign stays the same.
• To solve $-4y \geq 20$, divide by -4 (a negative number): $\frac{-4y}{-4} \leq \frac{20}{-4}$, so $y \leq -5$. The sign reverses.
• To solve $\frac{z}{-2} < 5$, multiply by -2 (a negative number): $-2(\frac{z}{-2}) > -2(5)$, so $z > -10$. The sign reverses.

Explanation

The game-changing rule! When you multiply or divide both sides by a positive number, nothing changes. But if you use a negative number, you MUST flip the inequality sign. For example, > becomes <.

Property

A linear inequality is an inequality in one variable that can be written in one of the following forms where $a$, $b$, and $c$ are real numbers and $a \neq 0$:

To solve, follow the same steps as for linear equations, but reverse the inequality sign when multiplying or dividing by a negative number.

Examples

• To solve $7x - 5 > 3x + 11$, subtract $3x$ from both sides to get $4x - 5 > 11$. Then add 5 to get $4x > 16$. Finally, divide by 4 to get $x > 4$.
• To solve $3(k - 2) \leq 12$, distribute to get $3k - 6 \leq 12$. Add 6 to both sides to get $3k \leq 18$. Divide by 3 to get $k \leq 6$.
• To solve $2y \geq 9y + 35$, subtract $9y$ from both sides to get $-7y \geq 35$. Divide by -7 and reverse the sign to get $y \leq -5$.

Explanation

Combine everything you know! Simplify each side, collect variables on one side, and then isolate the variable. Just keep an eye out for that one tricky step: multiplying or dividing by a negative number flips the inequality sign.

Property

When solving an inequality, if the variable is eliminated and the result is a true statement (like $-10 < 36$), the inequality is an identity and the solution is all real numbers, $(-\infty, \infty)$. If the result is a false statement (like $0 > 18$), the inequality is a contradiction and there is no solution.

Examples

• The inequality $5(x+2) > 5x+3$ simplifies to $5x+10 > 5x+3$, then $10 > 3$. This is true, so the solution is all real numbers, $(-\infty, \infty)$.
• The inequality $2(y-3) \leq 2y-10$ simplifies to $2y-6 \leq 2y-10$, then $-6 \leq -10$. This is false, so there is no solution.
• The inequality $4z - 1 - 4z < -1$ simplifies to $-1 < -1$. This is false because -1 is not less than itself, so there is no solution.

Explanation

Sometimes the variables completely cancel out! If you are left with a true statement like $5 < 10$, the answer is all real numbers. If you are left with a false statement like $5 > 10$, there is no solution.

Property

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some common phrases are 'is greater than' ($>$), 'is at least' ($\geq$), 'is less than' ($<$), and 'is at most' ($\leq$).

Examples

• 'Thirty less than a number $n$ is at least 50' translates to $n - 30 \geq 50$.
• 'Four times a number $y$ is no more than 24' translates to $4y \leq 24$.
• 'A number $p$ increased by 10 exceeds 25' translates to $p + 10 > 25$.

Explanation

Math has its own language. Learning keywords helps you translate from English to an inequality. 'Exceeds' means 'greater than,' while 'at most' means 'less than or equal to.' Pay close attention to these phrases.

Property

To solve applications with linear inequalities, first identify what you are looking for and assign a variable. Translate the problem into an inequality by representing the situation. Solve the inequality, and ensure the answer makes sense in the context of the problem, rounding if necessary.

Examples

• A grant is 3,000 dollars for science kits. Each kit costs 75 dollars. What is the maximum number of kits, $k$, you can buy? The inequality is $75k \leq 3000$. Solving gives $k \leq 40$. You can buy a maximum of 40 kits.
• Your phone bill is 25 dollars plus 0.15 dollars per text, $t$. To keep your bill no more than 40 dollars, the inequality is $25 + 0.15t \leq 40$. Solving gives $t \leq 100$. You can send at most 100 texts.
• To earn a profit of at least 500 dollars, revenue must exceed costs by that amount. If you sell items for 10 dollars each and expenses are 400 dollars, how many items, $b$, must you sell? The inequality is $10b - 400 \geq 500$. Solving gives $b \geq 90$ items.

Explanation

Use inequalities to solve real-world problems about budgets, profits, or limits. Set up an expression for the total cost or amount and compare it to the limit using an inequality sign. Then, solve for your variable.