Testing and Explaining Solutions of Inequalities | Grade 6 Math | LevelKite

You already know how to test a value in an equation. You put the value in place of the variable (the letter that stands for a number) and see whether the two sides come out equal.

Now you will run the same kind of test on a different statement. An inequality is a statement that compares two quantities with one of the symbols $<$ , $>$ , $\le$ or $\ge$ . An equation says its two sides are equal. An inequality says how one side compares with the other: less than, greater than, or one of these with "or equal to" added.

The table shows how to read each symbol.

The four inequality symbols

Symbol	How to read it	A true example	A false example
$<$	is less than	$3 < 5$	$5 < 5$
$>$	is greater than	$9 > 5$	$5 > 5$
$\le$	is less than or equal to	$5 \le 5$	$6 \le 5$
$\ge$	is greater than or equal to	$5 \ge 5$	$4 \ge 5$

Look at the inequality $x > 3$ , where $x$ stands for a number. On its own, it is neither true nor false. It becomes true or false once you choose a number for $x$ .

Choose $4$ . Is $4 > 3$ ? Yes, so the statement is true.
Choose $10$ . Is $10 > 3$ ? Yes. True again.
Choose $3$ . Is $3 > 3$ ? No. $3$ equals $3$ , so it is not greater than $3$ . False.
Choose $1$ . Is $1 > 3$ ? No. False.

Putting a number in place of the variable is called substituting. A number that makes the statement true is called a solution. So $4$ and $10$ are solutions of $x > 3$ , while $3$ and $1$ are not. Notice that this one inequality has more than one solution.

Key concept

A solution makes the statement true

A value is a solution of an inequality when substituting it for the variable gives a true comparison. When the comparison is false, the value is not a solution.

You can trust the test only when you:

put the same value in place of every copy of the variable,
work out each side on its own, following the order of operations (for example, multiplying before adding),
read the symbol exactly, including whether equal sides count.

One test settles one value. It tells you nothing about any other value.

Here is the test, one move at a time.

Substitute. Replace the variable with the value you are testing.
Work out each side. Find the value of the left side, then the value of the right side. This does not change the statement, because a side such as $3(5) + 2$ and its value, $17$ , are the same number.
Compare. Read the two values with the inequality's own symbol. Is the comparison true?
Decide. True means the value is a solution. False means it is not.

Worked example

Test one value

Is $5$ a solution of $3x + 2 > 14$ ? Here $x$ stands for a number.

Substitute 5 for x on the left side, because you judge a solution by what the inequality says when x is that number. 3x means 3 times x, so it becomes 3 times 5. The right side has no x, so it stays 14.
$3x + 2 \quad \text{becomes} \quad 3(5) + 2$
Work out the left side. Multiply first and then add, because the order of operations puts multiplication before addition.
$3(5) + 2 = 15 + 2 = 17$
Compare the two values with the symbol from the inequality. It asks whether the left side is greater than the right side. 17 is greater than 14, so the comparison is true.
$17 > 14 \quad \text{(true)}$
Decide. A true comparison means the tested value is a solution.

Answer

Yes, $5$ is a solution of $3x + 2 > 14$ , because substituting $5$ gives $17 > 14$ , which is true.

Worked example

The variable on both sides

Is $2$ a solution of $4x - 1 \ge 2x + 5$ ? Here $x$ stands for a number.

Substitute 2 for x in both places. The letter x means the same number everywhere in the statement, so every copy gets the same value.
$4(2) - 1 \quad \text{and} \quad 2(2) + 5$
Work out the left side on its own. Multiply first, then subtract, following the order of operations.
$4(2) - 1 = 8 - 1 = 7$
Work out the right side on its own. Multiply first, then add.
$2(2) + 5 = 4 + 5 = 9$
Compare with the symbol from the inequality. It asks whether the left side is greater than or equal to the right side. 7 is less than 9, so it is neither greater than 9 nor equal to 9. The comparison is false, so 2 is not a solution.
$7 \ge 9 \quad \text{(false)}$

Answer

No, $2$ is not a solution of $4x - 1 \ge 2x + 5$ , because substituting $2$ gives $7 \ge 9$ , which is false.

Many problems give you a set of values to choose from. A set is a collection of numbers, listed inside braces, such as $\{1, 2, 3, 4, 5\}$ . The question becomes: which values from this set, if any, make the statement true?

To answer, test each value in the set, one at a time. The answer may be several values, one value, or none. Only values from the given set can be part of the answer.

The table below tests every value in $\{1, 2, 3, 4, 5\}$ in three statements, where $x$ stands for a number: the equation $2x + 1 = 7$ , the inequality $2x + 1 > 7$ , and the inequality $2x + 1 \ge 7$ . All three have the same left side. So you find the value of $2x + 1$ once for each value of $x$ , and then ask three questions about it.

Every value in the set, tested in three statements

Value tested	Left side	Equal to 7?	Greater than 7?	Greater than or equal to 7?
$1$	$2(1) + 1 = 3$	no	no	no
$2$	$2(2) + 1 = 5$	no	no	no
$3$	$2(3) + 1 = 7$	yes	no	yes
$4$	$2(4) + 1 = 9$	no	yes	yes
$5$	$2(5) + 1 = 11$	no	yes	yes

Read down each question column to collect the solutions from this set.

$2x + 1 = 7$ is true only for $3$ .
$2x + 1 > 7$ is true for $4$ and $5$ .
$2x + 1 \ge 7$ is true for $3$ , $4$ and $5$ .

The number $6$ also makes $2x + 1 > 7$ true, since $2(6) + 1 = 13$ and $13 > 7$ . But $6$ is not in this set, so it is not part of the answer here.

An inequality can have more than one solution in a set. So keep testing after you find the first value that works.

Worked example

Find the solutions in a set

Which values in the set $\{1, 2.5, 4, 6\}$ make $4x - 3 < 13$ true? Here $x$ stands for a number.

Test 1. Substitute 1 for x, multiply, then subtract. The left side is 1, and 1 is less than 13, so the comparison is true.
$4(1) - 3 = 4 - 3 = 1, \quad 1 < 13 \quad \text{(true)}$
Test 2.5. Four times 2.5 is 10, and 10 minus 3 is 7. 7 is less than 13, so the comparison is true.
$4(2.5) - 3 = 10 - 3 = 7, \quad 7 < 13 \quad \text{(true)}$
Test 4. The left side comes out to exactly 13. The symbol asks for less than, and 13 is not less than 13, so the comparison is false.
$4(4) - 3 = 16 - 3 = 13, \quad 13 < 13 \quad \text{(false)}$
Test 6. The left side is 21, which is greater than 13, not less than 13. The comparison is false.
$4(6) - 3 = 24 - 3 = 21, \quad 21 < 13 \quad \text{(false)}$
Collect the values that gave a true comparison. You have tested every value in the set, so the list is complete for this set.

Answer

From the set $\{1, 2.5, 4, 6\}$ , the values $1$ and $2.5$ make $4x - 3 < 13$ true. The values $4$ and $6$ do not.

Deciding is only half of the job. You also need to say why. A complete explanation has three parts:

the value you substituted,
the comparison you got after working out each side,
whether that comparison is true or false, using what the symbol means.

Look back at the answers to the first two examples. Each one has these three parts.

"It looks big enough" is not an explanation. Neither is "a nearby number worked." In this lesson, the reason always comes from the substitution result for that exact value.

Worked example

Explain both decisions

Explain why $8$ is not a solution of $20 - 2x \ge 6$ and why $7$ is a solution. Here $x$ stands for a number.

Substitute 8 for x and work out the left side. Multiply first, then subtract. The right side has no x, so it stays 6.
$20 - 2(8) = 20 - 16 = 4$
Compare. The symbol asks whether the left side is greater than or equal to 6. 4 is less than 6, so the comparison is false.
$4 \ge 6 \quad \text{(false)}$
Substitute 7 for x and work out the left side the same way.
$20 - 2(7) = 20 - 14 = 6$
Compare. The left side equals 6. The symbol includes "or equal to", so equal sides make the comparison true.
$6 \ge 6 \quad \text{(true)}$
Write each explanation with its three parts: the value, the comparison you got, and whether it is true. Notice that the larger value failed and the smaller one passed, so the size of the value alone could not have told you the answer.

Answer

$8$ is not a solution of $20 - 2x \ge 6$ because substituting it gives $4 \ge 6$ , which is false: $4$ is less than $6$ . $7$ is a solution because substituting it gives $6 \ge 6$ , which is true: equal sides count for $\ge$ .

The takeaway. A solution of an inequality is a value that makes it true. To find out whether a value is a solution, substitute it for every copy of the variable, work out each side, and read the comparison with the inequality's exact symbol. Equal sides make an inequality true only for $\le$ and $\ge$ . When you are given a set, test every value in it, because each check settles only the value you tested. When you explain, give the comparison you got and say whether it is true.