Grade by Grade
C. Algebra · Patterns, Integers, Equations, Inequalities, Coding and Modelling

Inequalities with Multiple Terms: Solve, Verify and Graph

C2.4 A1About 50 minutes
Learning goals
  • I can simplify an inequality by combining like terms.
  • I can solve an inequality by finding the boundary point and testing values.
  • I can graph the solutions on a number line, with dots, or with an open or closed circle and a shaded line.
  • I can verify my solutions and explain what they mean in real life.

Words to know

inequality
A statement that two expressions are not equal, using <, >, ≤ or ≥.
≤ and ≥
≤ means less than or equal to. ≥ means greater than or equal to.
boundary point
The number where the two sides would be equal. It separates solutions from non-solutions.
open circle
On a graph, shows that the boundary point is not a solution (for < or >).
closed circle
On a graph, a filled-in circle that shows the boundary point is a solution (for ≤ or ≥).
verify
Check a solution by substituting it into the original inequality.

Remember from Grade 6

In Grade 6 you solved inequalities with two operations and whole numbers up to 100, like 3n + 4 < 40. You found the boundary point by solving the matching equation, tested a number on each side, and graphed the whole-number solutions as dots on a number line.

In Grade 7 the inequalities have multiple terms, like 4x + 2x + 5 < 41, so you combine like terms first. You'll also learn a second way to graph: with an open or closed circle and a shaded line.

Let's start on the water

A canoe can safely carry at most 350 kg. Two adults (80 kg and 75 kg) are paddling, and they want to bring packs of 15 kg each. How many packs can they bring?

At most 350 means the total must be less than or equal to 350: 80 + 75 + 15p ≤ 350. There isn't just one answer. Lots of numbers of packs work. Let's find them all.

How to solve an inequality

  1. Simplify each side by combining like terms.
  2. Find the boundary point: solve the matching equation (change the inequality sign to =).
  3. Test a number on one side of the boundary to see which direction works. Testing is safer than guessing!
  4. Write the solution, like x < 6 or x ≥ 6.
  5. Graph the solutions.
  6. Verify with a number from your solution, and check that a number outside it does not work.
SignBoundary point included?Graph (any numbers)Graph (whole numbers only)
<noopen circle, shade leftdots below the boundary
>noopen circle, shade rightdots above the boundary
≤yesclosed circle, shade leftdots up to and including the boundary
≥yesclosed circle, shade rightdots from the boundary up
Shade left and right are for inequalities like x < 6 or x ≥ 6, where x is on the left. Always test to be sure.

Two kinds of graphs

If the variable can be any number (like a length or a mass), the solutions include decimals too. Then you draw a circle at the boundary point and shade the number line in the direction of the solutions. An arrow shows that the solutions keep going.

If the variable can only be a whole number (like people, packs or tickets), draw a dot on each whole-number solution.

Worked example

Example 1: Combine, then solve

Solve 4x + 2x + 5 < 41 and graph the solutions.

  1. Simplify: 4x + 2x = 6x, so 6x + 5 < 41.
  2. Boundary: solve 6x + 5 = 41. Subtract 5: 6x = 36. Divide by 6: x = 6.
  3. Test x = 0: 6(0) + 5 = 5, and 5 < 41 is true. So the solutions are on the same side as 0: x < 6.
  4. Verify: x = 5 gives 30 + 5 = 35 < 41 ✓. x = 7 gives 42 + 5 = 47, which is not < 41 ✓ (not a solution). x = 6 gives 41, which is not < 41, so the circle is open.

x < 6. Graph: open circle at 6, shaded to the left.

x < 60123456789101112
Graph of x < 6: an open circle at 6 and a shaded line going left, with an arrow. Every number less than 6, including decimals like 5.9, is a solution.
Worked example

Example 2: Greater than or equal to

Solve 3n + 2n − 4 ≥ 26 and graph the solutions.

  1. Simplify: 5n − 4 ≥ 26.
  2. Boundary: 5n − 4 = 26. Add 4: 5n = 30. n = 6.
  3. Test n = 10: 50 − 4 = 46, and 46 ≥ 26 is true. So the solutions are on the same side as 10: n ≥ 6.
  4. Verify: n = 6 gives 26 ≥ 26 ✓ (included, so the circle is closed). n = 5 gives 21, not ≥ 26 ✓.

n ≥ 6. Graph: closed circle at 6, shaded to the right.

n ≥ 60123456789101112
Graph of n ≥ 6: a closed (filled) circle at 6 and a shaded line going right, with an arrow.
Worked example

Example 3: The canoe

Solve 80 + 75 + 15p ≤ 350, where p is the number of packs.

  1. Simplify: 80 + 75 = 155, so 155 + 15p ≤ 350.
  2. Boundary: 155 + 15p = 350. Subtract 155: 15p = 195. Divide by 15: p = 13.
  3. Test p = 0: 155 ≤ 350 is true. So the solutions are p ≤ 13.
  4. Packs come in whole numbers, so the solutions are 0, 1, 2, ..., 13.
  5. Verify: p = 13 gives 155 + 195 = 350 ≤ 350 ✓. p = 14 gives 365, too heavy ✓.

They can bring at most 13 packs (0 to 13).

0123456789101112131415
Whole-number solutions of 155 + 15p ≤ 350: dots on 0 to 13, including 13. No dots on 14 or 15.
Worked example

Example 4: When the variable is subtracted

Solve 50 − 2x − 3x ≥ 10.

  1. Simplify: 50 − 5x ≥ 10.
  2. Boundary: 50 − 5x = 10. The 5x must be 40, so x = 8.
  3. Test x = 0: 50 ≥ 10 is true. 0 is less than 8, so the solutions are x ≤ 8.
  4. Notice: the sign is ≥, but the solutions are less than the boundary. That's because subtracting a bigger 5x makes the left side smaller. Testing caught it!

x ≤ 8. Graph: closed circle at 8, shaded to the left.

x ≤ 80123456789101112
Graph of x ≤ 8: a closed circle at 8 and a shaded line going left.

Always test

Don't just follow the direction of the sign. Pick an easy number (0 is often easiest), put it into the original inequality and see if it's true. Then you know which side of the boundary to shade.

Just for fun

Why is an inequality so easygoing?

It never insists on just one answer: 0, 1, 2, ... any of them will do!

Hands-on activity

Pack the backpack

You need: a kitchen or bathroom scale, a backpack, and some items like books or water bottles. Ask a grown-up first.

  1. Weigh the empty backpack. Choose a limit, like 5 kg.
  2. Weigh one book (or bottle). Suppose all your books are about that mass.
  3. Write an inequality: backpack + (book mass) × b ≤ limit. If the backpack is 1 kg and each book is 0.5 kg, it's 1 + 0.5b ≤ 5.
  4. Solve it: find the boundary, test, and write the whole-number solutions. (For this example: b ≤ 8, so 0 to 8 books.)
  5. Verify with the scale: pack the largest number of books and weigh it. Then try one more book.
  6. Talk about it: why might real books not match your answer exactly? (They don't all have the same mass.)

Your turn

1

Solve 2x + 3x < 30.

2

Solve 4n + 7 > 31.

3

Solve 3y + y + 2 ≤ 22.

4

Describe the graph of x ≥ 4.

5

Is 9 a solution of 2a + 4a − 5 > 50?

6

List the whole-number solutions of 3k + 2k < 20.

7

You have $50 (all prices made up). Skate sharpening costs $8. You buy c hot chocolates at $3 and the same number of cookies at $2. Write and solve an inequality to find how many of each you can buy.

More practice: warm-up

1

Is 3 a solution of x + 4 < 10?

2

What is the boundary point of 2x > 18?

3

Does x < 5 include 5?

4

For ≤, is the circle open or closed?

5

Write "t is at most 20" as an inequality.

More practice: on your own

1

Solve 6m − 2m + 3 ≥ 27 and describe the graph. Verify with one value.

2

Solve 40 − 3x − x > 12.

3

An elevator can carry at most 900 kg. A worker (80 kg) and a cart (40 kg) ride with boxes of 30 kg each. How many boxes can they take?

4

Zoe says the solution of 5x + 3x < 48 is x > 6. Test a value to show she is wrong, and give the correct solution.

5

A graph shows an open circle at 3 and shading to the right. Write the inequality.

Challenge

1

For a school fundraiser, the class sells n bags of popcorn at $2 and n juice boxes at $3. Supplies cost $15. They want to make at least $120 after paying for supplies. Write and solve an inequality.

2

Find all whole numbers that make both inequalities true: 3x + 2 > 14 and 2x + x < 21.

3

In 100 − 4x > 60, the sign is >, but the solutions are x < 10. Explain why.

Check your understanding

Quiz

1.

Solve 3x + 3x < 42.

2.

Which graph shows x ≥ 2?

3.

What is the boundary point of 5n + 2n + 6 ≤ 55?

4.

Is 4 a solution of 2y + 3y − 1 > 19?

5.

Solve 30 − 2x − x ≥ 9.

6.

A van holds at most 8 people. The driver and a coach are already in it. Which inequality shows how many players p can ride?

For parents and teachers

Big idea: Grade 7 students solve inequalities that involve multiple terms and whole numbers, and verify and graph the solutions (C2.4). An inequality has many solutions. The method: combine like terms, find the boundary point by solving the matching equation, test a value to choose the side, graph, and verify.

Common difficulties:

  • Open vs closed circle. < and > use an open circle; ≤ and ≥ use a closed one.
  • Shading by the shape of the sign without testing. In 50 − 5x ≥ 10, the solutions are x ≤ 8. Testing 0 makes this clear.
  • Real-life whole numbers. c ≤ 8.4 means at most 8 drinks.
  • Words: "at most" means ≤, "at least" means ≥.

How to help: Look for limits in daily life: elevator capacity signs, baggage limits, a budget for an outing. Ask "How many could we...?" and solve together. This lesson builds A1 skills: reasoning, checking with examples and connecting math to real decisions.