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C2.4 · Solve, verify, and graph whole-number inequalities

Learn to solve, verify, and graph whole-number inequalities through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Algebra

Ontario Grade 7 Mathematics — C2.4

A comparison can allow one number, or it can allow several numbers. For example, a rule that says “no more than five players” allows any whole-number player count from zero through five. Finding all the allowed counts is solving the condition. Checking a count against the rule is verifying it. A number line can show the results clearly. In this lesson, you will use whole-number values and comparison words to find, check, and graph solutions.

What you will learn

Recognize when a condition compares whole-number amounts.
Find whole numbers that make a condition true by testing them.
Verify a proposed solution by checking the condition.
Graph whole-number solutions on a number line.

1. Compare whole-number amounts

First, recall whole numbers:

0, 1, 2, 3, 4, \ldots

. They are zero and the counting numbers. When a question asks for whole-number solutions, do not include fractions or negative numbers.

An inequality is a statement that compares amounts. The symbol

<

means “is less than.” The symbol

>

means “is greater than.” For example,

3<7

is true because three is less than seven. The comparison

7>3

is also true. The same two amounts are compared, but their order is switched.

Some comparison words include the limit. “No more than five” means five or any smaller amount. “At most five” and “up to five” have the same meaning. “Fewer than five” means an amount smaller than five, but not five itself. “At least five” includes five and larger amounts. These phrases help you decide whether the limit is allowed.

A solution is a value that makes a condition true. For the rule “no more than five players,” the possible whole-number counts are CAD 0, 1, 2, 3, 4, 5. Each count follows the rule. Six players does not. The collection of values that work is called the solution set.

A whole number is zero or a counting number.
“Fewer than” and “greater than” leave out the stated limit.
“No more than” and “at least” include the stated limit.

2. Find and verify solutions

To solve a whole-number condition, test whole-number values and keep the ones that make the condition true. When the amount counts objects, start at zero because a count cannot be negative. Work in order so you do not miss a value.

For example, consider the condition “a number plus three is less than eight.” Try zero: zero plus three is three, and three is less than eight. Try one: one plus three is four, so that works too. Continue through the whole numbers. Four plus three is seven, which is still less than eight. Five plus three is eight, which is not less than eight. So the values from zero through four work, and five and larger values do not.

To verify means to check a proposed value in the original condition. For instance, check four: adding three gives seven, which is less than eight. Check five: adding three gives eight, which is not less than eight. This confirms that four works and five does not.

For a count involving equal groups, use the situation to check values. Suppose each stack contains three books and the total must be no more than eighteen books. Five stacks contain fifteen books; six contain eighteen; seven contain twenty-one. Five and six meet the limit, but seven does not. More stacks would mean more books, so they would not meet the limit either.

A boundary is the value at the edge between those that work and those that do not. Check the boundary directly. Also check a nearby whole number when it helps show whether the limit is included. The words in the condition matter: exactly eighteen is allowed by “no more than eighteen,” but exactly eight does not satisfy “less than eight.”

Test whole-number values in order and keep those that make the condition true.
Verify a value by checking it in the original condition.
Check the boundary and a nearby value to decide whether the boundary is included.

3. Graph the solutions

A number line places numbers in order. For a whole-number solution graph, mark the whole-number values that work. A filled dot means that value is included. An open dot means that value is not included.

For the condition “a number plus three is less than eight,” the solutions are zero through four. Mark those whole numbers. Five is the boundary value that does not work, so show it as an open endpoint. Do not mark fractions between the whole numbers when the question asks for whole-number solutions.

For a condition such as “no more than six stacks,” mark zero through six. The dot at six is filled because six is allowed. The graph should agree with the list: each marked number works, and every whole-number solution shown in the range is marked.

Some conditions have no greatest whole-number solution. For example, a count of at least six includes six and every larger whole number. On a number line, use a filled dot at six and show that the solutions continue to the right. The point at six is included, even though the solutions continue.

Use a filled dot when the boundary value works.
Use an open endpoint when the boundary value does not work.
Make sure the graph and the list show the same whole-number solutions.

4. Use a careful routine

Read the situation and identify what is being counted or compared. Notice key words such as “fewer than,” “no more than,” and “at least.” They tell you whether the boundary is included.

Test whole-number values in a sensible order. Write down the comparison for each value near the boundary. Keep a value only when the comparison is true. Use the situation to rule out counts that cannot make sense, such as a negative number of objects.

Verify the boundary and a nearby value in the original condition. Then list the solutions or graph them. Check that your words, list, and graph agree. The graph is a picture of the same solutions, not a different answer.

Read the comparison words carefully.
Check proposed values in the original condition.
Make the list and graph agree.

Worked example

A strict upper limit

A number plus three must be less than eight. Find all whole-number values that work, verify the boundary, and graph the solutions.

Whole-number solutions

Filled marks show solutions from 0 to 4. The open endpoint at 5 is not a solution.

Test values near the edge
Start at zero and test whole numbers in order. Four plus three is seven, which is less than eight. Five plus three is eight, which is not less than eight. Larger values will give totals greater than eight, so they do not work either.
$4+3=7<8;\quad 5+3=8$
List and graph
Every whole number from zero through four works. Five is not included. The graph marks zero through four and shows five as an open endpoint.
$0,1,2,3,4$

Answer: The whole-number solutions are CAD 0, 1, 2, 3, 4.

Check: Four gives a true comparison:

4+3<8

. Five gives a false comparison because

5+3=8

, which is not less than eight.

Worked example

A limit that includes its boundary

Each stack has three books. There can be no more than eighteen books. Find the whole-number stack counts that work, verify the limit, and graph the solutions.

Stack counts that meet the limit

Mark whole-number counts 0 through 6. The filled endpoint at 6 is included.

Check stacks near the limit
Multiply the number of stacks by three to find the number of books. Five stacks give fifteen books, and six give eighteen. Seven stacks give twenty-one, which is too many. The limit of eighteen is included because the rule says “no more than.”
$3\times5=15;\quad 3\times6=18;\quad 3\times7=21$
State the solutions
Zero through six stacks meet the rule. Any greater stack count contains more than eighteen books, so it does not work. The endpoint at six is included on the graph.
$0,1,2,3,4,5,6$

Answer: The possible whole-number stack counts are CAD 0, 1, 2, 3, 4, 5, 6.

Check: Six stacks give exactly eighteen books, which meets the limit. Seven stacks give twenty-one books, which exceeds it.

Worked example

A lower limit

A whole-number count must be at least six. Find the values that work, check six and the number just below it, and graph the solutions.

Whole numbers at least six

The filled mark at 6 is included. Larger whole-number solutions continue to the right.

Check the boundary
“At least six” includes six and all larger amounts. Six meets the condition. Five is below six, so it does not meet the condition.
$6\geq6;\quad 5<6$
List and graph
The solutions are six and every larger whole number. Use a filled dot at six and show the solutions continuing to the right.
$6,7,8,\ldots$

Answer: The whole-number solutions are CAD 6, 7, 8 and every larger whole number.

Check: Six is included because it equals the stated limit. Five is not included because it is less than six.

Common mistakes and how to avoid them

Treating “no more than” as if the limit is not allowed.

Correction: The phrase “no more than” includes the stated limit. Check the limit in the condition.

Including the boundary for “less than.”

Correction: Test the boundary. A value equal to the limit is not less than the limit.

Including fractions when only whole-number solutions are requested.

Correction: Keep only whole numbers that make the condition true.

Marking a value as a solution without checking it.

Correction: Put the value into the original condition and decide whether the comparison is true.

Lesson summary

An inequality compares amounts. Comparison words tell whether the limit is included.
Solve by testing whole-number values and keeping the ones that make the condition true.
Verify proposed values in the original condition, especially near the boundary.
Graph the whole-number solutions with filled or open endpoints as appropriate.

Check your understanding

Question 1

A count must be at least four. Which values are solutions?

0,1,2,3
0,1,2,3,4
$4,5,6,\ldots$
0,1,2,3,4,5

Show answer and explanation

4,5,6,\ldots

“At least four” includes four and every larger whole number.

Question 2

Which whole numbers make “a number plus two is no more than six” true?

0,1,2,3
0,1,2,3,4
$4,5,6,\ldots$
0,1,2,3,4,5

Show answer and explanation

0,1,2,3,4

Four plus two is six, so four is included. Five plus two is seven, so five and larger values fail.

Question 3

Does five satisfy “a number minus one is greater than two”?

Yes, because five is greater than two.
Yes, because five minus one is two.
No, because five minus one is four, which is greater than two.
No, because five is a whole number.

Show answer and explanation

No, because five minus one is four, which is greater than two.

Check the original comparison: five minus one is four, and four is greater than two. Therefore five does satisfy the condition. The correct option is the one stating that it does.

Key terms

Whole number: Zero or a counting number, such as $0, 1, 2, 3, \ldots$ .
Inequality: A statement that compares amounts using signs such as $<$ and $>$ , or comparison words.
Solution: A value that makes a condition true.
Verify: Check a proposed value in the original condition.
Boundary: The value at the edge between values that work and values that do not.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 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.

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