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

Inequalities: Solving, Checking and Graphing

C2.3 A1About 40 minutes

Learning goals

I can tell the difference between an equation and an inequality.
I can solve inequalities with adding and subtracting, using whole numbers up to 20.
I can verify a solution by testing it.
I can graph the solutions of an inequality on a number line.

Words to know

inequality
A math sentence that says two sides are not equal. It uses < (is less than) or > (is greater than).

<
is less than. 3 < 8 means 3 is less than 8.

>
is greater than. 9 > 4 means 9 is greater than 4.

solutions of an inequality
All the numbers that make an inequality true. There is usually more than one!

whole numbers
The numbers 0, 1, 2, 3, 4 and so on (no fractions or decimals).

graph the solutions
Mark every solution with a dot on a number line.

Remember

You have used < and > to compare numbers: 12 > 7 (12 is greater than 7) and 5 < 9 (5 is less than 9). The open side of the sign faces the bigger number.

In the last two lessons, every equation had one solution. Today you will meet inequalities, which can have many solutions!

Let's start with a story

Jayden is playing a board game. He has 7 points. To win, he needs more than 15 points. How many more points, p, could he score to win?

We write: 7 + p > 15 (7 plus p is greater than 15).

This is an inequality. It does not say the two sides are equal. It says one side is greater.

How to solve an inequality

Find the balance point. Pretend it is an equation: 7 + p = 15. The solution is p = 8. At 8, Jayden has exactly 15 points. That is not more than 15, so 8 is not a solution.
Test a number on each side of the balance point. Try p = 9: 7 + 9 = 16. Is 16 > 15? Yes ✔ Try p = 7: 7 + 7 = 14. Is 14 > 15? No ✘
Decide. Numbers greater than 8 work. So p > 8. Jayden could score 9, 10, 11, 12, ... more points.
Graph the solutions on a number line.

Solutions of 7 + p > 15: every whole number greater than 8. The dots on 9 to 20 are solutions, and the solutions keep going past 20 (21, 22, ...).

Verify

Pick any dot and check it. Try p = 12: 7 + 12 = 19, and 19 > 15 ✔. Try a number with no dot, like 5: 7 + 5 = 12, and 12 > 15 is false ✘. The graph is correct.

Equation	Inequality
Has an = sign	Has a < or > sign
n + 5 = 12 has one solution: 7	n + 5 < 12 has many solutions: 0, 1, 2, 3, 4, 5, 6
The two sides have the same value	One side is less than (or greater than) the other

Equations and inequalities.

Worked example

Example 1: A less-than inequality

Solve n + 5 < 12, where n is a whole number. Graph the solutions.

Balance point: n + 5 = 12, so n = 7. At 7, 7 + 5 = 12, and 12 < 12 is false. So 7 is not a solution.
Test 6: 6 + 5 = 11. Is 11 < 12? Yes ✔
Test 8: 8 + 5 = 13. Is 13 < 12? No ✘
So the solutions are whole numbers less than 7: 0, 1, 2, 3, 4, 5, 6.

n < 7, so n can be 0, 1, 2, 3, 4, 5 or 6. There are 7 solutions.

Solutions of n + 5 < 12: dots on 0, 1, 2, 3, 4, 5 and 6.

Worked example

Example 2: A subtraction inequality

Solve n − 3 > 10, using whole numbers up to 20. Graph the solutions.

Balance point: n − 3 = 10, so n = 10 + 3 = 13. At 13, 13 − 3 = 10, and 10 > 10 is false.
Test 14: 14 − 3 = 11. Is 11 > 10? Yes ✔
Test 12: 12 − 3 = 9. Is 9 > 10? No ✘
Solutions: whole numbers greater than 13.

n > 13: n can be 14, 15, 16, 17, 18, 19, 20 (and more past 20).

Solutions of n − 3 > 10 up to 20: dots on 14 to 20.

Worked example

Example 3: An inequality from a story

Ella has already spent $9 on a gift. She wants to buy a book too, but she needs to spend less than $20 altogether. How much can the book cost, in whole dollars?

Let b be the cost of the book. Total spent: 9 + b. It must be less than 20.
Inequality: 9 + b < 20.
Balance point: 9 + b = 20, so b = 11. That makes exactly $20, which is not less than 20.
Test 10: 9 + 10 = 19 < 20 ✔. Test 12: 9 + 12 = 21 < 20? No ✘.

b < 11: the book can cost $10 or less ($1, $2, ..., $10).

Worked example

Example 4: Is it a solution?

Which of 4, 9, 15 are solutions of 20 − n > 8?

Test 4: 20 − 4 = 16. Is 16 > 8? Yes ✔
Test 9: 20 − 9 = 11. Is 11 > 8? Yes ✔
Test 15: 20 − 15 = 5. Is 5 > 8? No ✘

4 and 9 are solutions. 15 is not. (Notice: when n is subtracted, smaller values of n make the left side bigger. That's why testing is so important!)

Three steps to remember

Balance point: solve it as if it had an = sign.
Test a number just above and just below the balance point.
Graph and verify: put dots on all the solutions, then check one dot and one empty spot.

The balance point itself is never a solution for < or >, because the two sides would be equal.

Just for fun

Why are the signs < and > always so hungry?

Their mouths are always open toward the bigger number!

Hands-on activity

Number line inequality game

You need: a long strip of paper or tape on the floor, a marker, 21 small paper squares or coins, and a die (or slips of paper numbered 1 to 6).

Draw a number line from 0 to 20 on the strip, with the numbers spread out.
Your grown-up writes an inequality, like n + 4 < 11.
Find the balance point (n = 7). Test a number above and below.
Put a coin or square on every solution on your number line. Count them.
Your grown-up points to one covered number and one uncovered number. Verify both out loud.
Take turns: roll the die to choose the number added (n + 🎲 < 15), and make up new inequalities for each other.

What to look for: for n + 4 < 11 the coins are on 0 to 6, and your child can show why 7 is not a solution.

Grown-up note: keep small items like these away from babies and toddlers.

Your turn

Is n + 2 > 9 an equation or an inequality?

Is n = 3 a solution of n + 6 < 10?

Solve n + 4 < 9 (whole numbers). List all the solutions.

Solve n + 10 > 18, using whole numbers up to 20.

Solve n − 5 < 6 (whole numbers). Remember n − 5 must make sense, so n is at least 5.

Describe the graph of the solutions of n + 1 < 4.

A sled can carry less than 20 kg. A box on it is 12 kg. Write an inequality for the extra mass, m, you could add (whole kilograms).

More practice: warm-up

Read 6 < 10 out loud.

True or false: 14 > 17

Is n = 2 a solution of n + 3 < 8?

Is n = 5 a solution of n + 3 < 8?

Find the balance point of n + 7 > 12.

More practice: on your own

Solve n + 8 < 15 and graph the solutions.

Solve n − 6 > 9 for whole numbers up to 20.

Which of 3, 8, 12 are solutions of n + 9 > 17?

Raj has 6 hockey cards. He wants more than 14 cards. How many more, c, does he need? Write and solve an inequality (up to 20).

Kim says the solutions of n + 5 < 10 are 0, 1, 2, 3, 4 and 5. Is she right?

Challenge

Solve 15 − n > 9 for whole numbers up to 15. Be careful!

Find all whole numbers that make both n + 3 > 7 and n + 3 < 12 true.

Write a story for n + 4 < 10 and give all its solutions.

Check your understanding

Quiz

Which is an inequality?

Which number is a solution of n + 6 > 13?

What are all the whole-number solutions of n + 2 < 5?

A number line has dots on 12, 13, 14, ..., 20. Which inequality could it show?

Why is 7 not a solution of n + 3 < 10?

How many whole-number solutions does n + 4 < 8 have?

For parents and teachers

Big idea: The Ontario curriculum (C2.3) asks Grade 4 students to solve inequalities that involve addition and subtraction of whole numbers up to 20, and verify and graph the solutions. The big shift from equations is that an inequality usually has many solutions, shown as dots on a number line.

Common difficulties:

Including the balance point. For n + 5 < 12, 7 is not a solution because 12 < 12 is false.
Forgetting 0. 0 is a whole number, so it is often a solution of a less-than inequality.
Mixing up < and >. The open side faces the bigger number. Reading the sign out loud ("is less than") helps.
Inequalities where n is subtracted (15 − n > 9). The smaller numbers work here. Testing a number on each side catches this.

How to help: Use real limits: "The bag can hold fewer than 15 apples and has 9. How many more could fit?" Draw a 0 to 20 number line and mark solutions with coins. Ask your child to verify one solution and one non-solution (Strand A: reasoning and proving).