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

Inequalities with Two Operations: Solve, Verify and Graph

C2.4 A1About 40 minutes

Learning goals

I can find the boundary point of an inequality with two operations by solving an equation.
I can test numbers to decide which side of the boundary point is the solution.
I can graph the whole-number solutions on a number line.
I can verify my solutions and use inequalities to solve real problems.

Words to know

inequality
A statement that two amounts are not equal, using <, >, ≤ or ≥.

boundary point
The number that makes both sides equal. It marks the edge of the solutions.

≤
Less than or equal to. n ≤ 5 means n can be 5 or anything less.

≥
Greater than or equal to. n ≥ 5 means n can be 5 or anything greater.

solution of an inequality
Every value that makes the inequality true. There are usually many.

Remember from Grade 5

In Grade 5 you solved inequalities with one operation and whole numbers up to 50, like 4 × n < 28. You found the boundary point, tested a number on each side, graphed the solutions with dots on a number line, and verified.

In Grade 6 the inequalities have two operations, like 3n + 4 < 40, and use whole numbers up to 100. You'll also meet two new symbols: ≤ and ≥.

Let's start at the arena

The Grade 6 class wants to book ice time for a skating party. The arena charges a $10 booking fee plus $6 for each student's skate rental. The class has less than $70 to spend. How many students can rent skates?

6n + 10 < 70

Boundary point: solve 6n + 10 = 70. Undo + 10: 6n = 60. Undo × 6: n = 10. At 10 the cost is exactly $70, which is not less than 70. So 10 is not a solution.
Test below: n = 9. 6 × 9 + 10 = 64. Is 64 < 70? Yes ✔
Test above: n = 11. 6 × 11 + 10 = 76. Is 76 < 70? No ✘
Decide: the solutions are whole numbers less than 10: 0, 1, 2, ..., 9.

Rows of black rental ice skates on wooden shelves next to an indoor ice rink. — Each pair of rental skates adds the same cost, on top of one booking fee. That is why the cost can be written as 6n + 10. AI-generated image

Solutions of 6n + 10 < 70: dots on 0 to 9. There is no dot on the boundary point, 10.

Verify: check one dot and one spot with no dot. n = 4: 24 + 10 = 34 < 70 ✔. n = 12 (no dot): 72 + 10 = 82 < 70 ✘. The graph is correct. So up to 9 students can rent skates.

Symbol	Read it as	Is the boundary point a solution?
<	is less than	No
>	is greater than	No
≤	is less than or equal to	Yes
≥	is greater than or equal to	Yes

The line under ≤ and ≥ is like half of an equal sign: it means equal is allowed.

Five steps for any inequality

Boundary: change the sign to = and solve (undo + or − first, then × or ÷).
Included? With < or >, the boundary is not a solution. With ≤ or ≥, it is.
Test one number below and one above the boundary.
Graph a dot on every whole-number solution.
Verify with a dot and with a spot that has no dot.

Worked example

Example 1: Two operations

Solve 3n + 4 < 40. Graph the whole-number solutions.

Boundary: 3n + 4 = 40. 3n = 36, so n = 12. With <, 12 is not a solution.
Test 11: 33 + 4 = 37. Is 37 < 40? Yes ✔
Test 13: 39 + 4 = 43. Is 43 < 40? No ✘
Solutions: whole numbers less than 12.

n < 12: 0, 1, 2, ..., 11.

Solutions of 3n + 4 < 40: dots on 0 to 11.

Worked example

Example 2: Subtraction and greater than

Solve 2n − 6 > 30, using whole numbers up to 100.

Boundary: 2n − 6 = 30. Undo − 6 by adding 6: 2n = 36, so n = 18. Not included.
Test 20: 40 − 6 = 34. Is 34 > 30? Yes ✔
Test 15: 30 − 6 = 24. Is 24 > 30? No ✘
Solutions: whole numbers greater than 18, up to 100.

n > 18: 19, 20, 21, ..., 100.

Part of the graph of 2n − 6 > 30: dots on 19, 20, 21 and onward. The dots continue all the way to 100.

Worked example

Example 3: Less than or equal to

Solve 5n + 15 ≤ 90.

Boundary: 5n + 15 = 90. 5n = 75, so n = 15. With ≤, 15 is a solution: 75 + 15 = 90, and 90 ≤ 90 is true.
Test 14: 70 + 15 = 85 ≤ 90 ✔. Test 16: 80 + 15 = 95 ≤ 90 ✘.
Solutions: whole numbers less than or equal to 15.

n ≤ 15: 0, 1, 2, ..., 15. This time the boundary point gets a dot.

Solutions of 5n + 15 ≤ 90: dots on 0 to 15, including 15.

Worked example

Example 4: The variable is subtracted

Solve 100 − 3n > 64.

Boundary: 100 − 3n = 64. What do we take from 100 to get 64? 36. So 3n = 36 and n = 12. Not included.
Test 10: 100 − 30 = 70. Is 70 > 64? Yes ✔
Test 14: 100 − 42 = 58. Is 58 > 64? No ✘
Careful! The sign is >, but the solutions are below 12, because taking away more makes the left side smaller.

n < 12: 0, 1, 2, ..., 11. Always test. Don't just follow the direction of the sign.

Worked example

Example 5: Dividing

Solve n ÷ 4 + 3 > 8, using whole numbers up to 100.

Boundary: n ÷ 4 + 3 = 8. Undo + 3: n ÷ 4 = 5. Undo ÷ 4: n = 5 × 4 = 20. Not included.
Test 24: 24 ÷ 4 + 3 = 6 + 3 = 9. Is 9 > 8? Yes ✔
Test 16: 16 ÷ 4 + 3 = 4 + 3 = 7. Is 7 > 8? No ✘
What about 21? 21 ÷ 4 = 5.25, and 5.25 + 3 = 8.25 > 8 ✔. So every whole number above 20 works.

n > 20: 21, 22, 23, ..., 100.

Worked example

Example 6: A fundraiser

The class sells popcorn for $3 a bag. They already have $25. They want at least $100. How many bags must they sell?

"At least $100" means $100 or more: 3c + 25 ≥ 100.
Boundary: 3c + 25 = 100. 3c = 75, so c = 25. With ≥, 25 is included.
Test 26: 78 + 25 = 103 ≥ 100 ✔. Test 24: 72 + 25 = 97 ≥ 100 ✘.

c ≥ 25. They must sell at least 25 bags.

Just for fun

Memory trick: The open side of < or > always faces the bigger amount, like a hungry alligator going for the bigger snack. In 64 < 70, the alligator is chomping toward the 70. (Real alligators have not been consulted.)

Hands-on activity

Budget number line

You need: a grocery flyer (or the prices below), a long strip of paper or painter's tape on the floor, a marker, and about 20 coins or buttons.

Make a number line from 0 to 15 on the paper or tape.
The problem: you have at most $50. You must buy one bag of oranges for $8, and then as many boxes of crackers at $6 each as you can. Write the inequality: 6n + 8 ≤ 50.
Find the boundary point. Is it included?
Test a number on each side. Then put a coin on every number that is a solution.
Verify: pick one coin and one empty spot and check them.
Change the problem: what if the crackers cost $7? Move the coins.

What to look for: 6n + 8 = 50 gives 6n = 42, so n = 7, and 7 is included (≤). Coins on 0 to 7. With $7 crackers: 7n + 8 ≤ 50, 7n = 42 at the boundary, so n = 6: coins on 0 to 6.

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

Your turn

Solve 2n + 5 < 25.

Solve 4n − 8 > 20, using whole numbers up to 100.

Solve 3n + 6 ≤ 36.

Is n = 9 a solution of 5n + 3 > 50?

Solve n ÷ 2 + 1 < 6 for whole numbers. Which numbers get dots?

A cart has a mass of 20 kg. Each box on it is 8 kg. A sign says the total must be no more than 100 kg. Write and solve an inequality for the number of boxes, b.

What is the boundary point of 7n + 2 > 51?

More practice: warm-up

Read n > 6 in words.

Is 4 a solution of n + 3 < 10?

Is 5 a solution of n ≤ 5?

What is n if 2n = 18?

True or false: 12 ≥ 12.

More practice: on your own

Solve 6n + 9 > 63, using whole numbers up to 100.

Solve 8n + 4 ≤ 60.

Solve 100 − 4n > 60.

Popcorn sells for $3 a bag. A team has $40 and wants more than $91. Write and solve an inequality.

Pat says 10 is a solution of 3n + 1 < 31. Is Pat right?

Challenge

Find every whole number that makes both 2n + 3 > 15 and 3n − 2 < 40 true.

Write an inequality with two operations whose whole-number solutions are exactly 0, 1, 2, 3, 4, 5.

In the arena story (6n + 10 < 70), the booking fee goes up to $22. Now how many students can rent skates? How many fewer than before?

Check your understanding

Quiz

What is the boundary point of 3n + 5 < 32?

What are the solutions of 2n + 4 > 20?

Which number is a solution of 5n − 3 ≤ 42?

What does ≥ mean?

When you graph n ≤ 6, does 6 get a dot?

What are the solutions of 100 − 2n > 80?

For parents and teachers

Big idea: Grade 6 students solve inequalities that involve two operations and whole numbers up to 100, and verify and graph the solutions (C2.4). The method from Grade 5 still works: find the boundary point by solving the matching equation, test a number on each side, graph, and verify.

Common difficulties:

Including the boundary point with < or > (or leaving it out with ≤ or ≥).
Following the sign's direction without testing. In 100 − 3n > 64, the solutions are less than 12. Testing catches this every time.
Undoing in the wrong order when finding the boundary (divide before subtracting).

How to help: Use budgets: "We have at most $50. One item is $8 and the rest cost $6 each. How many can we buy?" Draw a number line on paper or tape one on the floor and place coins on the solutions. Social-emotional learning (A1): testing numbers is how mathematicians check their thinking. Praise testing, especially when it finds a mistake.