Is It a Solution? Testing Values in Inequalities | Grade 7 Math | LevelKite

You already know how to check a solution of an equation. To test $x = 5$ in $x + 4 = 9$ , you replace the variable $x$ with $5$ and see that $5 + 4 = 9$ is true. (A variable is a letter that stands for a number.)

An inequality is a statement that compares two quantities with one of the symbols $<$ , $>$ , $\le$ or $\ge$ instead of $=$ . Like an equation, it has a left side and a right side. The table below shows what each symbol means. Notice which symbols still give a true statement when the two sides are equal.

The four inequality symbols

Symbol	Read it as	True when both sides are equal?	Examples
$<$	is less than	No	$3 < 5$ is true; $5 < 5$ is false
$>$	is greater than	No	$7 > 5$ is true; $5 > 5$ is false
$\le$	is less than or equal to	Yes	$3 \le 5$ is true; $5 \le 5$ is true
$\ge$	is greater than or equal to	Yes	$7 \ge 5$ is true; $5 \ge 5$ is true

A solution of an inequality is a value of the variable that makes the inequality a true statement. (A real situation can also limit which values make sense. You will see that in the last example.)

Take $x + 4 > 9$ . It says that a number plus $4$ is greater than $9$ . To test a value, you substitute it: replace $x$ with the value, compute the left side, and compare the result with $9$ .

The equation $x + 4 = 9$ has exactly one solution, $x = 5$ . The inequality $x + 4 > 9$ has many solutions. The table below tests four values.

Testing four values in x + 4 > 9

Value of x	Left side: x + 4	Compare with 9	Is the value a solution?
$2$	$2 + 4 = 6$	$6 > 9$ is false	No
$5$	$5 + 4 = 9$	$9 > 9$ is false	No
$5.5$	$5.5 + 4 = 9.5$	$9.5 > 9$ is true	Yes
$6$	$6 + 4 = 10$	$10 > 9$ is true	Yes

Key concept

A solution makes the inequality true

To test a value in an inequality:

Substitute the value for the variable.
Compute each side.
Compare the results with the symbol exactly as written.

If the comparison is false, the value is not a solution. If it is true, the value is a solution, unless a real situation rules that value out.

Keep three conditions in mind:

A boundary value makes the two sides equal. It makes the inequality true for $\le$ or $\ge$ , but false for $<$ or $>$ .
The symbol reverses (for example, $>$ becomes $<$ ) when you multiply or divide both sides of an inequality by a negative number. Substituting does neither, so the symbol stays as written.
A real situation may allow only whole numbers ( $0, 1, 2, 3, \ldots$ ) or only nonnegative values (zero or greater).

Worked example

Test two values

Is $x = 4$ a solution of $3x + 2 \ge 14$ ? Is $x = 3$ a solution?

Substitute 4 for x on the left side, because a solution must make the inequality true when it replaces the variable. Multiply first, then add.
$3(4) + 2 = 12 + 2 = 14$
Compare the left side, 14, with the right side, 14, using the symbol as written. The two sides are equal, so 4 is a boundary value. The symbol means greater than or equal to, so equal sides make a true statement.
$14 \ge 14 \text{ is true}$
Now substitute 3 for x and compute the left side in the same way.
$3(3) + 2 = 9 + 2 = 11$
Compare 11 with 14 using the same symbol. Because 11 is 3 less than 14, it is not greater than 14 and not equal to 14. The statement is false.
$11 \ge 14 \text{ is false}$

Answer

The value $x = 4$ is a solution of $3x + 2 \ge 14$ , and the value $x = 3$ is not a solution.

Worked example

Test values when negative numbers appear

Which of the values $x = -5$ , $x = -4$ and $x = 2$ are solutions of $-3x > 12$ ?

Substitute negative 5 for x. A negative number times a negative number is positive, so the left side is 15. Keep the symbol as written, because substituting only computes each side. Since 15 is greater than 12, the statement is true.
$-3(-5) = 15, \quad 15 > 12 \text{ is true}$
Substitute negative 4 for x. The left side is 12, which equals the right side, so negative 4 is a boundary value. This symbol means greater than only. It does not include equal to, so equal sides make a false statement.
$-3(-4) = 12, \quad 12 > 12 \text{ is false}$
Substitute 2 for x. A negative number times a positive number is negative, so the left side is negative 6. Every negative number is less than every positive number, so negative 6 is not greater than 12. The statement is false.
$-3(2) = -6, \quad -6 > 12 \text{ is false}$

Answer

Only $x = -5$ is a solution of $-3x > 12$ . The values $x = -4$ and $x = 2$ are not solutions.

Worked example

When the situation limits the values

Each ride at a fair uses $4$ tickets, and you have $30$ tickets. Let $r$ be the number of rides you take. Then $r$ rides use $4r$ tickets. You cannot use more tickets than you have, so $4r \le 30$ . Which of the values $r = 7$ , $r = 8$ , $r = 7.5$ and $r = -2$ are solutions in this situation?

First decide which values the situation allows. The variable r counts rides, so it must be a whole number: 0, 1, 2, 3 and so on. You cannot take part of a ride or a negative number of rides.
Substitute 7 for r and multiply to find the tickets used. You use 28 tickets, and 28 is less than 30, so the statement is true. The situation allows 7, because 7 is a whole number.
$4(7) = 28, \quad 28 \le 30 \text{ is true}$
Substitute 8 for r. You would use 32 tickets, and 32 is greater than 30, so the statement is false. The situation allows 8, but 8 does not make the inequality true.
$4(8) = 32, \quad 32 \le 30 \text{ is false}$
Substitute 7.5 for r. Splitting 7.5 into 7 and 0.5 shows that the left side is 30. It equals the right side, and the symbol means less than or equal to, so the statement is true. But 7.5 is not a whole number, and you cannot take half a ride, so the situation rules it out.
$4(7.5) = 4(7) + 4(0.5) = 28 + 2 = 30, \quad 30 \le 30 \text{ is true}$
Substitute negative 2 for r. A positive number times a negative number is negative, so the left side is negative 8. A negative number is less than a positive number, so the statement is true. But a number of rides cannot be negative, so the situation rules out negative 2 as well.
$4(-2) = -8, \quad -8 \le 30 \text{ is true}$

Answer

In this situation, only $r = 7$ is a solution. The value $r = 8$ makes $4r \le 30$ false. The values $r = 7.5$ and $r = -2$ make it true, but the situation allows only whole numbers of rides, so they are not solutions here.

Takeaway: A value is a solution of an inequality when substituting it makes the inequality a true statement. Substitute, compute each side, and compare with the symbol exactly as written. Remember:

A boundary value makes the two sides equal. It makes the inequality true for $\le$ and $\ge$ , but false for $<$ and $>$ .
The symbol reverses when you multiply or divide both sides by a negative number. Substituting does neither.
A real situation may allow only whole numbers or only nonnegative values.

Next, you will graph the solutions of an inequality on a number line and write inequalities from number line graphs.