Solve, verify, and graph integer inequalities

Ontario Grade 8 Mathematics

Algebra

A Grade 8 lesson on finding and showing integer answers

An equation says that two amounts are equal. An inequality compares amounts that may not be equal. For example, a rule might say that a whole-number score must be at least 4. More than one integer can meet that rule. In this lesson, you will use inverse operations to find all integer solutions, check that your answer works, and show the solutions on a number line.

What you will learn

Explain what an inequality tells us.
Solve an integer inequality by keeping both sides balanced.
Verify a proposed solution by checking it in the original inequality.
Graph the integer solutions on a number line.

1. What an inequality says

An integer is a whole number, its opposite, or zero. Examples include

-3

,

0

, and

8

. For this lesson, the solutions we graph are integers.

An inequality compares two quantities. The symbol

<

means “is less than.” The symbol

>

means “is greater than.” The symbol

\leq

means “is less than or equal to,” and

\geq

means “is greater than or equal to.” The bar means the boundary value is included.

Imagine a rule that says a player needs at least 4 points. A score of 4 meets the rule, and so do scores above 4. The word “at least” includes 4, so we use

\geq

. An inequality can therefore have several integer solutions, not just one.

A number line helps show which integers work. A filled dot means the boundary number is included. An open dot means it is not included. Shade or draw an arrow toward the integers that satisfy the rule.

a < b,\ a > b,\ a \leq b,\ a \geq b

“At least” and “no less than” mean $\geq$ .
“At most” and “no greater than” mean $\leq$ .
Filled dot: included boundary. Open dot: excluded boundary.

2. Solve by keeping both sides balanced

To solve an inequality, find the values of the variable that make the comparison true. A variable is a letter that stands for a number. Use inverse operations to undo operations around the variable. Addition and subtraction undo each other. Multiplication and division undo each other.

Keep an inequality balanced by doing the same operation to both sides. For example, if you subtract 3 from the left side, subtract 3 from the right side too. This keeps the comparison fair.

When you multiply or divide both sides by a negative number, reverse the inequality sign. For example,

<

becomes

>

. This is necessary because multiplying negative numbers can reverse their order:

-2 < 1

, but multiplying each number by

-1

gives

2 > -1

.

After solving, remember that the question asks for integer solutions. A statement such as

n > 2

means the integer answers start at 3, not at 2. A statement such as

n \leq 2

includes 2 and every smaller integer.

-a < b\ \Longrightarrow\ a > -b

Use the same addition, subtraction, multiplication, or division on both sides.
Reverse the inequality sign when multiplying or dividing both sides by a negative number.
Read the final inequality carefully to identify the integer answers.

3. Verify and graph

To verify a solution, substitute a proposed integer into the original inequality. Then check whether the comparison is true. Testing a boundary value is especially useful: it can reveal whether the boundary should be included.

For instance, if solving gives

x > 1

, test

x=1

in the original inequality to see whether the boundary works. Then test an integer on each side of 1. The graph should agree with these checks.

On a number line, place the dot at the boundary. Use a filled dot for

\leq

or

\geq

, and an open dot for

<

or

>

. Shade toward the solutions. If the solutions are integers, only the integer marks represent answers.

Check three things before finishing: the algebra keeps the inequality balanced, any sign reversal is handled, and the graph includes exactly the integers that make the original statement true.

x < c\ or\ x > c

Substitution checks whether a proposed value makes the original comparison true.
The boundary dot matches whether equality is allowed.
The graph and the solved inequality must describe the same integer values.

Worked example

Undo addition

Solve

x+5\leq 2

. Verify the boundary and graph the integer solutions.

Integer solutions to x less than or equal to negative 3

Filled dot at negative 3 shows that the boundary is included; the arrow points to smaller integers.

Undo the addition
The variable has 5 added to it. Subtract 5 from both sides so the variable is by itself. Subtracting the same amount keeps the comparison balanced.
$x+5-5\leq 2-5$
State the solution
The result says that $x$ is less than or equal to $-3$ . The boundary is included because the inequality uses “or equal to.” The integer solutions are $-3$ and every integer to its left.
$x\leq -3$
Verify the boundary
Substitute $-3$ into the original inequality. The left side becomes 2, so the comparison is true. The boundary belongs in the solution set.
$-3+5=2\leq 2$

Answer: The integer solutions are

-3,-4,-5,\ldots

.

Check: The number line has a filled dot at

-3

and extends left.

Worked example

Undo subtraction and verify a boundary

Solve

y-4>-1

. Verify whether the boundary value works, then graph the integer solutions.

Integer solutions to y greater than 3

Open boundary at 3; the integer solutions begin at 4 and continue right.

Undo subtraction
Add 4 to both sides to undo subtracting 4. The inequality sign stays the same because we are adding, not multiplying or dividing by a negative number.
$y-4+4>-1+4$
Read the integer solutions
The result is $y>3$ . Since 3 is not included, the first integer solution is 4. Every larger integer also works.
$y>3$
Check the boundary
Substitute 3 in the original statement. It gives $3-4>-1$ , which is false because $-1$ is not greater than $-1$ . So 3 is correctly excluded. The next integer, 4, gives a true comparison.
$4-4>-1$

Answer: The integer solutions are

4,5,6,\ldots

.

Check: Use an open dot at 3 and extend the graph to the right.

Worked example

Divide by a negative number

Solve

-2m\geq 6

. Verify one solution and one value that does not work. Graph the integer solutions.

Integer solutions to m less than or equal to negative 3

The filled boundary is included; solutions extend toward smaller integers.

Undo multiplication
Divide both sides by $-2$ to leave $m$ by itself. Because the divisor is negative, reverse the inequality sign.
$m\leq -3$
Verify a solution
Try $m=-4$ . The original left side is 8, and 8 is greater than or equal to 6. So $-4$ works.
$-2(-4)=8\geq 6$
Check a value that does not work
Try $m=-2$ . The original left side is 4, which is not greater than or equal to 6. So $-2$ does not work. The graph should stop at $-3$ and include that boundary.
$-2(-2)=4\not\geq 6$

Answer: The integer solutions are

-3,-4,-5,\ldots

.

Check: A filled dot at

-3

and shading to the left matches

m\leq -3

.

Common mistakes and how to avoid them

Changing only one side when using an inverse operation.

Correction: Do the same operation to both sides to keep the comparison balanced.

Keeping the same inequality sign when dividing by a negative number.

Correction: Reverse the sign when multiplying or dividing both sides by a negative number.

Using a filled dot for a strict inequality such as

x>3

.

Correction: Use an open dot when the boundary is not included.

Including a boundary value without checking whether equality is allowed.

Correction: Use a filled dot only for

\leq

or

\geq

, and verify the boundary in the original inequality.

Lesson summary

An inequality compares quantities and may have many integer solutions.
Use inverse operations on both sides to isolate the variable.
Reverse the inequality sign when multiplying or dividing by a negative number.
Verify values in the original inequality.
Graph the boundary with the correct dot and extend toward the integer solutions.

Check your understanding

Question 1

Solve

a+2<5

. Which statement gives the integer solutions?

$a<3$ ; the integers are $2,1,0,\ldots$
$a\leq 3$ ; the integers include 3
$a>3$ ; the integers begin at 4
correctIndex":0,

Show answer and explanation

a<3

; the integers are

2,1,0,\ldots

Subtract 2 from both sides to get

a<3

. Since the boundary is excluded, the integer solutions are 2 and smaller.

Question 2

Solve

-3b<9

. Which result is correct?

$b<-3$
$b>-3$
$b\geq -3$
correctIndex":1,

Show answer and explanation

b>-3

Divide both sides by

-3

and reverse the sign. This gives

b>-3

.

Key terms

Integer: A whole number, its opposite, or zero.
Inequality: A statement that compares quantities using a sign such as $<$ , $>$ , $\leq$ , or $\geq$ .
Variable: A letter that stands for a number.
Verify: Check a proposed answer by substituting it into the original inequality.
Boundary: The number that marks where the graph changes between included and excluded values.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C2.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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C2.4 · Solve, verify, and graph integer inequalities

Ontario Grade 8 Mathematics

A Grade 8 lesson on finding and showing integer answers

What you will learn

1. What an inequality says

2. Solve by keeping both sides balanced

3. Verify and graph

Undo addition

Undo subtraction and verify a boundary

Divide by a negative number

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Grade 8 Math

About this lesson and its review