C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Inequalities with One Operation: Solve, Verify and Graph

C2.4 A1About 40 minutes

Learning goals

I can write an inequality for a story.
I can solve inequalities with one operation (+, −, × or ÷) using whole numbers up to 50.
I can verify solutions by testing numbers.
I can graph the solutions on a number line.

Words to know

inequality
A math sentence that compares two amounts that are not equal, using < or >.

<
is less than. 4 × 5 < 30 means 4 × 5 is less than 30.

>
is greater than. 45 > 40 means 45 is greater than 40.

boundary point
The number that would make the two sides equal. It splits the solutions from the non-solutions. (In Grade 4 we called it the balance point.)

solutions of an inequality
All the numbers that make the inequality true.

Remember from Grade 4

In Grade 4 you solved inequalities like 7 + p > 15 with numbers up to 20. You found the balance point (where both sides are equal), tested a number on each side, and put dots on a number line for all the solutions.

In Grade 5 you'll use numbers up to 50, and your inequalities can use any one operation: adding, subtracting, multiplying or dividing. We'll call the balance point the boundary point, because it marks the edge of the solutions.

Let's start with a story

The sledding hill has a toboggan that can carry less than 28 kg of gear at one time. Each bag of gear has a mass of 4 kg. How many bags, n, can go on the toboggan?

4 × n < 28

Boundary point: solve 4 × n = 28. n = 7. Seven bags is exactly 28 kg, which is not less than 28. So 7 is not a solution.
Test a number below: n = 6. 4 × 6 = 24. Is 24 < 28? Yes ✔
Test a number above: n = 8. 4 × 8 = 32. Is 32 < 28? No ✘
Decide: the solutions are whole numbers less than 7: 0, 1, 2, 3, 4, 5, 6.

A wooden sled loaded with canvas gear bags tied down with rope at the top of a snowy hill. — Bags of gear tied onto a sled at the top of a hill. The total mass must stay less than 28 kg, so only some numbers of bags are solutions. AI-generated image

Solutions of 4 × n < 28: dots on 0, 1, 2, 3, 4, 5 and 6.

Verify: check one dot and one empty spot. n = 5: 4 × 5 = 20 < 28 ✔. n = 10 (no dot): 40 < 28 ✘. The graph is correct.

Worked example

Example 1: An addition inequality up to 50

Solve n + 18 > 40, using whole numbers up to 50. Graph the solutions.

Boundary point: n + 18 = 40, so n = 22. At 22, both sides equal 40, so 22 is not a solution.
Test 23: 23 + 18 = 41. Is 41 > 40? Yes ✔
Test 21: 21 + 18 = 39. Is 39 > 40? No ✘
Solutions: whole numbers greater than 22, up to 50.

n > 22: 23, 24, 25, ..., 50.

Solutions of n + 18 > 40 up to 50: dots on every whole number from 23 to 50.

Worked example

Example 2: The variable is subtracted

Solve 50 − n < 12, using whole numbers up to 50.

Boundary point: 50 − n = 12, so n = 50 − 12 = 38.
Test 39: 50 − 39 = 11. Is 11 < 12? Yes ✔
Test 37: 50 − 37 = 13. Is 13 < 12? No ✘
When n is subtracted, a bigger n makes the left side smaller. So the solutions are greater than 38, even though the sign is <.

n > 38: 39, 40, 41, ..., 50. Always test. Don't just follow the direction of the sign!

Solutions of 50 − n < 12: dots on 39 to 50.

Worked example

Example 3: When the boundary point isn't a whole number

Movie tickets cost $6 each. Kiran wants to spend less than $40. How many tickets, t, can he buy?

Inequality: 6 × t < 40.
Boundary point: 6 × t = 40. There is no whole number that works: 6 × 6 = 36 and 6 × 7 = 42. The boundary is between 6 and 7.
Test 6: 36 < 40 ✔. Test 7: 42 < 40 ✘.

t can be 0, 1, 2, 3, 4, 5 or 6. Kiran can buy up to 6 tickets.

Worked example

Example 4: A division inequality

Solve n ÷ 5 > 6, using whole numbers up to 50.

Boundary point: n ÷ 5 = 6, so n = 6 × 5 = 30. 30 ÷ 5 is exactly 6, not more than 6.
Test 31: 31 ÷ 5 = 6 R1. That's 6 and a bit, which is more than 6 ✔
Test 29: 29 ÷ 5 = 5 R4. That's 5 and a bit, which is less than 6 ✘
Bigger numbers divided by 5 give bigger answers.

n > 30: 31, 32, ..., 50.

Inequality	Boundary point	Solutions (whole numbers up to 50)
4 × n < 28	7	0 to 6
n + 18 > 40	22	23 to 50
50 − n < 12	38	39 to 50
6 × t < 40	between 6 and 7	0 to 6
n ÷ 5 > 6	30	31 to 50

A summary of the examples.

Boundary, test, graph, verify

Boundary: solve as if it had an = sign.
Test a whole number just below and just above.
Graph: dots on every solution.
Verify: check one dot and one empty spot.

The boundary point is never a solution for < or >, because there the two sides are equal.

Just for fun

In the toboggan story, n = 0 is a solution too: 4 × 0 = 0, and 0 < 28. Zero bags of gear is a perfectly good choice if all you want to carry down the hill is yourself!

Hands-on activity

Inequality number line game

You need: a strip of paper about 1 m long (or tape on the floor), a marker, about 30 coins or paper squares, and a die.

Draw a number line from 0 to 50. Mark every number, and write the label on every 5th one (0, 5, 10, ...).
Your grown-up writes an inequality, like 5 × n < 35.
Find the boundary point (7). Test 6 and 8.
Put a coin on every solution (0 to 6). Count the solutions.
Your grown-up points to one covered and one uncovered number. Verify both out loud.
Take turns: roll the die to choose a number, then make an inequality for each other, like 🎲 × n > 30 or n + 🎲 < 20.

What to look for: your child finds the boundary point first and tests both sides. For 5 × n < 35, the coins go on 0 to 6.

Grown-up note: keep coins away from babies and toddlers.

Your turn

Is n = 8 a solution of 5 × n > 35?

Solve 3 × n < 18 (whole numbers). List the solutions.

Solve n + 25 > 40, up to 50.

Solve n − 9 < 20 (whole numbers, n at least 9 so that n − 9 makes sense).

Solve n ÷ 4 > 8, up to 50.

Describe the graph of the solutions of 7 × n < 21.

A canoe can carry less than 45 kg of gear. A tent has a mass of 12 kg. Write and solve an inequality for the other gear, g, you can add (whole kilograms).

More practice: warm-up

Read 24 < 30 out loud.

True or false: 6 × 7 > 40

Is n = 5 a solution of n + 10 < 15?

Find the boundary point of 2 × n > 30.

Is n = 4 a solution of 9 × n < 40?

More practice: on your own

Solve 6 × n > 42, up to 50, and describe the graph.

Solve 40 − n > 25 (whole numbers up to 40).

Solve 8 × n < 50 (whole numbers).

Which of 12, 20, 27 are solutions of n ÷ 3 < 7?

Juice boxes come in packs of 4. The class needs more than 30 juice boxes. Write and solve an inequality for the number of packs, p.

Challenge

Solve n + n > 37, up to 50.

Write an inequality whose whole-number solutions are exactly 0, 1, 2, 3, 4 and that uses multiplication.

Ella says the solutions of 45 − n < 30 are 0 to 14, because the sign points to the small numbers. Is she right?

Check your understanding

Quiz

Which number is a solution of 7 × n < 35?

What is the boundary point of n + 27 > 50?

What are the solutions of n ÷ 2 > 20, up to 50?

Pencils cost $3 each. You want to spend less than $20. What is the greatest number of pencils you can buy?

What are the solutions of 30 − n < 10 (whole numbers up to 30)?

Why is the boundary point not a solution of an inequality with < or >?

For parents and teachers

Big idea: Grade 5 students solve inequalities that involve one operation and whole numbers up to 50, and verify and graph the solutions (C2.4). Unlike an equation, an inequality usually has many solutions. The reliable method is: find the boundary point, test a number on each side, graph, then verify.

Common difficulties:

Including the boundary point. Have your child substitute it and see that the two sides are equal.
Following the direction of the sign when the variable is subtracted (50 − n < 12). Testing one number on each side fixes this.
Boundary points that aren't whole numbers (6 × t < 40). Encourage testing the whole numbers on each side.

How to help: Use real limits: "The elevator holds fewer than 10 people," "We have less than $20 to spend." Ask your child to write the inequality and list every possible answer. Social-emotional learning (A1): inequalities ask children to check many numbers; praise persistence and careful checking.