Chapter 3 · Algebraic foundations

What Counts as a Solution of an Inequality?

Learn what it means for a number to be a solution of an inequality. Test values by substituting them, and watch for boundary values and limits set by a context.

12 min read

You already know how to solve an equation such as $x + 3 = 10$ , where $x$ stands for a number. Exactly one number makes it true: $x = 7$ , because $7 + 3 = 10$ .

An inequality is a statement that compares two expressions with one of these symbols instead of an equals sign:

$<$ means is less than
$>$ means is greater than
$\le$ means is less than or equal to
$\ge$ means is greater than or equal to

For example, $x + 3 > 10$ says that $x + 3$ is greater than $10$ . Which numbers make that statement true? That is the question for this lesson. Unless a situation limits the choices, $x$ can be any real number, so you may test positive numbers, negative numbers, zero, fractions and decimals.

A solution of an inequality is a value of the variable that makes the inequality a true statement. To test a value, substitute it into the inequality as written: replace the variable with the value, work out each side, and decide whether the comparison is true.

Test $x = 9$ in $x + 3 > 10$ . The left side is $9 + 3 = 12$ , and $12 > 10$ is true. So $9$ is a solution.
Test $x = 4$ . The left side is $4 + 3 = 7$ , and $7 > 10$ is false. So $4$ is not a solution.

The table below tests more values in the same way. Notice that this inequality has many solutions, while the equation $x + 3 = 10$ has only one.

Testing five values by substitution

Value tested	Left side	Statement	True or false?	Is it a solution?
$-2$	$-2 + 3 = 1$	$1 > 10$	False	No
$7$	$7 + 3 = 10$	$10 > 10$	False	No
$7.01$	$7.01 + 3 = 10.01$	$10.01 > 10$	True	Yes
$7.5$	$7.5 + 3 = 10.5$	$10.5 > 10$	True	Yes
$100$	$100 + 3 = 103$	$103 > 10$	True	Yes

Look at the row for $7$ . It makes the two sides equal, because $7 + 3 = 10$ . A value that makes the two sides of an inequality equal is called a boundary value. Each inequality with a variable in this lesson has exactly one boundary value. Here the boundary value is not a solution: $10 > 10$ is false, because a number is not greater than itself.

The symbol decides what happens at the boundary value:

$<$ and $>$ are strict. They are false when the two sides are equal, so a strict inequality excludes its boundary value.
$\le$ and $\ge$ are non-strict. They are true when the two sides are equal, so a non-strict inequality includes its boundary value, as long as the variable is allowed to take that value.

For example, $7$ is not a solution of $x + 3 > 10$ , but it is a solution of $x + 3 \ge 10$ , because $10 \ge 10$ is true.

Worked example

Testing values in a non-strict inequality

Which of the values $1$ , $3$ and $5$ are solutions of $2x + 1 \le 7$ ? Here $x$ can be any real number.

A solution must make the statement true, so substitute 1 for x and work out the left side, multiplying before adding. The result is 3, and 3 is less than 7. The statement is true, so 1 is a solution.
$2(1) + 1 = 2 + 1 = 3, \quad 3 \le 7 \text{ is true}$
Substitute 3 for x. The left side is 7, so the two sides are equal and 3 is the boundary value. The symbol is non-strict, so the statement is true when the sides are equal. That makes 3 a solution.
$2(3) + 1 = 6 + 1 = 7, \quad 7 \le 7 \text{ is true}$
Substitute 5 for x. The left side is 11, and 11 is greater than 7, so it is not less than or equal to 7. The statement is false, so 5 is not a solution.
$2(5) + 1 = 10 + 1 = 11, \quad 11 \le 7 \text{ is false}$

Answer

The values $1$ and $3$ are solutions of $2x + 1 \le 7$ , and $5$ is not a solution.

Worked example

Testing values in a strict inequality

Which of the values $-6$ , $-4$ and $0$ are solutions of $-3x > 12$ ? Here $x$ can be any real number.

Substitute negative 6 for x. A negative number times a negative number is positive, so the left side is 18. Since 18 is greater than 12, the statement is true and negative 6 is a solution.
$-3(-6) = 18, \quad 18 > 12 \text{ is true}$
Substitute negative 4 for x. The left side is 12, so the two sides are equal and negative 4 is the boundary value. The symbol is strict, and 12 is not greater than itself. The statement is false, so negative 4 is not a solution.
$-3(-4) = 12, \quad 12 > 12 \text{ is false}$
Substitute 0 for x. Any number times 0 is 0, and 0 is not greater than 12. The statement is false, so 0 is not a solution.
$-3(0) = 0, \quad 0 > 12 \text{ is false}$

Answer

Only $-6$ is a solution of $-3x > 12$ . The values $-4$ and $0$ are not solutions.

A common mistake. It is tempting to treat $-3x > 12$ like an equation: divide both sides by $-3$ , keep the same symbol, and write $x > -4$ . That would make $0$ a solution. But $-3(0) = 0$ , and $0 > 12$ is false.

The shortcut fails because multiplying or dividing both sides of an inequality by a negative number reverses its direction. Start with the true statement $18 > 12$ . Dividing both sides by the positive number $3$ keeps the order: $6 > 4$ . Taking the opposite of each side then swaps the order, because opposites are mirror images across zero: $-6 < -4$ . So dividing by $-3$ changes $>$ to $<$ , and the inequality with the same solutions is $x < -4$ .

Substituting into the original inequality avoids this step, so it is a safe test.

So far, the variable could be any real number. In a real situation, the context (the setting the numbers come from) can limit the values a variable is allowed to take. A count of people or objects, for instance, must be a whole number: $0$ , $1$ , $2$ , and so on. For a count, negative numbers and numbers between the whole numbers, such as $2.5$ , are not allowed.

In a context, a value is a solution only if it passes two tests: it is a value the variable is allowed to take, and it makes the inequality true.

Worked example

Solutions that must fit a context

A coach has $30$ players and wants to make teams of $4$ players each. Let $t$ be the number of teams the coach makes. The teams need $4t$ players, which cannot be more than $30$ , so $4t \le 30$ . Which of the values $7$ , $8$ , $7.5$ and $-2$ are solutions in this situation?

Decide which values t is allowed to take. It counts teams, so it must be a whole number: 0, 1, 2, and so on. A value is a solution here only if it is allowed and it makes the inequality true.
Test 7. It is a whole number, so it is allowed. Seven teams need 28 players, and 28 is less than 30. The statement is true, so 7 is a solution.
$4(7) = 28, \quad 28 \le 30 \text{ is true}$
Test 8. It is a whole number, so it is allowed. Eight teams need 32 players, and 32 is greater than 30. The statement is false, so 8 is not a solution.
$4(8) = 32, \quad 32 \le 30 \text{ is false}$
Test 7.5. The left side is 30, so the two sides are equal and 7.5 is the boundary value. The non-strict symbol makes the statement true. But the coach cannot make half a team, so 7.5 is not an allowed value and not a solution here.
$4(7.5) = 30, \quad 30 \le 30 \text{ is true}$
Test negative 2. The left side is negative 8, which is less than 30, so the statement is true. But a number of teams cannot be negative, so negative 2 is not an allowed value and not a solution here.
$4(-2) = -8, \quad -8 \le 30 \text{ is true}$

Answer

Of the four values, only $7$ is a solution in this situation, so the coach can make $7$ teams. The value $8$ makes the inequality false, and $7.5$ and $-2$ are not possible numbers of teams.

Key concept

What a solution of an inequality is

A solution of an inequality is a value of the variable that makes the inequality a true statement. An inequality can have many solutions. To test a value, substitute it into the original inequality and compare the two sides.

Three conditions apply:

Boundary. A strict inequality ( $<$ or $>$ ) is false at its boundary value, so it excludes that value. A non-strict inequality ( $\le$ or $\ge$ ) is true there, so it includes that value.
Negative numbers. Multiplying or dividing both sides of an inequality by a negative number reverses its direction. Substituting into the original inequality avoids that step.
Context. In a real situation, a solution must also be a value the variable is allowed to take.

Here is the idea to keep: a value is a solution of an inequality when it fits any limits set by the context and it makes the original inequality a true statement. A strict inequality excludes its boundary value, and a non-strict inequality includes it. Substituting tests the inequality as it is written. If you rewrite it by multiplying or dividing both sides by a negative number, its direction reverses.

An inequality such as $x + 3 > 10$ has too many solutions to list one by one. Next, you will show them all at once by graphing inequalities on number lines, and you will write inequalities from number lines.

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Graph and Write Inequalities on Number Lines

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