Solve Linear Inequalities

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality $x > 3$ true? Are you thinking, “$x$ could be 4”? That’s correct, but $x$ could be 5 too, or 20, or even 3.001. 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, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3.

The graph of $x \geq 3$ 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 $x = 3$ instead of a parenthesis.

Notice that the open parenthesis symbol, $($, shows that the endpoint of the inequality is not included. The closed bracket symbol, $[$, shows that the endpoint is included.

Example. Graph on the number line: (a) $x \leq 1$ (b) $x < 5$ (c) $x > -1$.

(a) This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at $x = 1$ to show that it is included.

(b) This means all numbers less than 5, but not including 5. We shade in all the numbers to the left of 5 and put a parenthesis at $x = 5$ to show it is not included.

(c) This means all numbers greater than $-1$, but not including $-1$. We shade in all the numbers to the right of $-1$, then put a parenthesis at $x = -1$ to show it is not included.

We can also represent inequalities using interval notation. As we saw above, the inequality $x > 3$ means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express $x > 3$ as $(3, \infty)$.

The symbol $\infty$ is read as “infinity.” It is not an actual number.

The inequality $x \leq 1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x \leq 1$ in interval notation as $(-\infty, 1]$. The symbol $-\infty$ is read as “negative infinity.”

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the shading? A parenthesis in interval notation matches an endpoint that is not included (graphed with an open circle or unfilled endpoint), and a bracket in interval notation matches an endpoint that is included (graphed with a filled endpoint).

Example. Graph on the number line and write in interval notation: (a) $x \geq -3$ (b) $x < 2.5$ (c) $x \leq -\tfrac{3}{5}$.

(a) Shade to the right of $-3$, and put a bracket at $-3$. In interval notation, this is $[-3, \infty)$.

(b) Shade to the left of $2.5$, and put a parenthesis at $2.5$. In interval notation, this is $(-\infty, 2.5)$.

(c) Shade to the left of $-\tfrac{3}{5}$, and put a bracket at $-\tfrac{3}{5}$. In interval notation, this is $\left(-\infty, -\tfrac{3}{5}\right]$.

Solve Inequalities Using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal. Similar properties hold true for inequalities.

For example, we know that $-4$ is less than $2$:

If we subtract 5 from both quantities, is the left side still less than the right side?

We get $-9$ on the left and $-3$ on the right:

The inequality sign stayed the same. Similarly, we could show that the inequality also stays the same for addition. This leads us to the Subtraction and Addition Properties of Inequality.

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality $x + 5 > 9$, we subtract 5 from both sides to isolate $x$:

Any number greater than 4 is a solution to this inequality.

Example. Solve the inequality $n - \tfrac{1}{2} \leq \tfrac{5}{8}$, graph the solution on the number line, and write the solution in interval notation.

Add $\tfrac{1}{2}$ to both sides of the inequality to isolate $n$:

Simplify:

The solution in interval notation is $\left(-\infty, \tfrac{9}{8}\right]$.

Solve Inequalities Using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0). Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider the true statement $10 < 15$. If we divide both sides by 5, we get $2 < 3$ — still true, and the inequality sign stayed the same. If we multiply both sides by 5, we get $50 < 75$ — again true, sign unchanged.

Now divide or multiply both sides of $10 < 15$ by $-5$ instead. Dividing gives $-2 \; ? \; -3$; since $-2 > -3$, the sign had to reverse to keep the statement true: $-2 > -3$. Multiplying gives $-50 \; ? \; -75$, and again the sign reverses: $-50 > -75$.

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Example. Solve the inequality $7y < 42$, graph the solution on the number line, and write the solution in interval notation.

Divide both sides of the inequality by 7. Since $7 > 0$, the inequality stays the same:

Simplify:

The solution in interval notation is $(-\infty, 6)$.

Example. Solve the inequality $-10a \geq 50$, graph the solution on the number line, and write the solution in interval notation.

Divide both sides of the inequality by $-10$. Since $-10 < 0$, the inequality reverses:

Simplify:

The solution in interval notation is $(-\infty, -5]$.

Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable on the left: $x > a$ has the same meaning as $a < x$. Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

Example. Solve the inequality $-20 < \tfrac{4}{5}u$, graph the solution on the number line, and write the solution in interval notation.

Multiply both sides of the inequality by $\tfrac{5}{4}$. Since $\tfrac{5}{4} > 0$, the inequality stays the same:

Simplify:

Rewrite the variable on the left:

The solution in interval notation is $(-25, \infty)$.

Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but we must be sure to pay close attention during multiplication or division.

Example. Solve the inequality $4m \leq 9m + 17$, graph the solution on the number line, and write the solution in interval notation.

Subtract $9m$ from both sides to collect the variables on the left:

Simplify:

Divide both sides by $-5$, and reverse the inequality since we’re dividing by a negative number:

Simplify:

The solution in interval notation is $\left[-\tfrac{17}{5}, \infty\right)$.

Example. Solve the inequality $8p + 3(p - 12) > 7p - 28$, graph the solution on the number line, and write the solution in interval notation.

Simplify each side as much as possible. Distribute:

Combine like terms:

Subtract $7p$ from both sides to collect the variables on the left, then simplify:

Add 36 to both sides to collect the constants on the right, then simplify:

Divide both sides by 4; the inequality stays the same since 4 is positive:

The solution in interval notation is $(2, \infty)$.

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity — the solution is all real numbers, $(-\infty, \infty)$. If the result is a false statement, we have a contradiction — there is no solution.

Example. Solve the inequality $8x - 2(5 - x) < 4(x + 9) + 6x$, graph the solution on the number line, and write the solution in interval notation.

Simplify each side as much as possible. Distribute:

Combine like terms:

Subtract $10x$ from both sides to collect the variables on the left:

Simplify:

The $x$’s are gone, and we have a true statement. The inequality is an identity — the solution is all real numbers, $(-\infty, \infty)$.

Example. Solve the inequality $\tfrac{1}{3}a - \tfrac{1}{8}a > \tfrac{5}{24}a + \tfrac{3}{4}$, graph the solution on the number line, and write the solution in interval notation.

Multiply both sides by the LCD, 24, to clear the fractions:

Simplify:

Combine like terms:

Subtract $5a$ from both sides to collect the variables on the left:

Simplify:

The statement is false! The inequality is a contradiction — there is no solution.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. Think about the phrase “at least” — what does it mean to be “at least 21 years old”? It means 21 or more. The phrase “at least” is the same as “greater than or equal to.”

Example. Translate and solve. Then write the solution in interval notation and graph on the number line: “Twelve times $c$ is no more than 96.”

Translate:

Solve — divide both sides by 12:

Simplify:

The solution in interval notation is $(-\infty, 8]$.

Example. Translate and solve. Then write the solution in interval notation and graph on the number line: “Thirty less than $x$ is at least 45.”

Translate:

Solve — add 30 to both sides:

Simplify:

The solution in interval notation is $[75, \infty)$.

Key terms

inequality — a mathematical statement comparing two expressions using $<$, $\leq$, $>$, or $\geq$, showing that one is smaller than, smaller than or equal to, larger than, or larger than or equal to the other. interval notation — a way of describing the solution set of an inequality using parentheses (for endpoints not included) and brackets (for endpoints included), together with $\infty$ or $-\infty$ where the solution set has no upper or lower end. identity (inequality) — an inequality that is true for every real number, with solution $(-\infty, \infty)$. contradiction (inequality) — an inequality with no solution, because no real number makes it true.

This section is adapted from Elementary Algebra 2e, Section 2.7: Solve Linear Inequalities by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the number-line figures as accessible inline graphics and the phrase-to-symbol reference as a markdown table; omitted the Be Prepared quiz, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.