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C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Solving and Graphing Inequalities with Integers

C2.4 A1About 50 minutes
Learning goals
  • I can solve inequalities that involve integers, including negative coefficients.
  • I can explain why the inequality sign reverses when I multiply or divide both sides by a negative number.
  • I can verify solutions by testing values, and graph solutions on a number line.

Words to know

inequality
A statement that two expressions are not equal, using <, >, ≤ or ≥.
boundary point
The number where the solutions start or stop. You find it by solving the matching equation.
open circle
Shows that the boundary point is not a solution (used with < and >).
closed circle
Shows that the boundary point is a solution (used with ≤ and ≥).
reverse the inequality
Turn the sign around (< becomes >, ≤ becomes ≥). You must do this when you multiply or divide both sides by a negative number.

Remember from Grade 7

In Grade 7 you solved inequalities with multiple terms, like 4x + 2x + 5 < 41. You found the boundary point, tested a number to choose the direction, and graphed the solutions with an open or closed circle and a shaded line.

In Grade 8 the inequalities involve integers: negative numbers, negative solutions, and negative coefficients. One new rule comes with that, and it's an important one.

Let's start with a freezer

A freezer is switched on. Its temperature is −6°C and it drops 3°C each hour until it reaches its setting. After how many hours is it at −18°C or colder?

"−18 or colder" means less than or equal to −18:

−6 − 3h ≤ −18

  • Add 6 to both sides: −3h ≤ −12.
  • Divide both sides by −3... and here's the twist: reverse the sign: h ≥ 4.

Test it: h = 5 gives −6 − 15 = −21, and −21 ≤ −18 ✓ (colder). h = 2 gives −12, and −12 ≤ −18 is false ✓. So the freezer is at −18°C or colder after 4 hours or more.

h ≥ 4012345678910
Closed circle at 4 (4 is included), shaded to the right.

Why does the sign reverse?

We know 2 < 5. Multiply both sides by −1: we get −2 and −5. On a number line, −2 is to the right of −5, so −2 > −5. Multiplying by a negative flips numbers to the other side of 0 and reverses their order. So the inequality sign must reverse too.

Adding, subtracting, or multiplying and dividing by a positive number never reverses the sign.

2 < 5, but −2 > −5−6−5−4−3−2−1012345625−2−5
2 is left of 5, so 2 < 5. Their opposites switch order: −2 is right of −5, so −2 > −5.

Steps for solving an inequality

  1. Simplify each side.
  2. Undo adding or subtracting (both sides).
  3. Undo multiplying or dividing (both sides). If you multiply or divide by a negative, reverse the sign.
  4. Test a value from your answer and one from outside it.
  5. Graph: open circle for < or >, closed circle for ≤ or ≥, and shade the solutions. If only integers make sense (people, goals, tickets), mark dots instead.
Worked example

Example 1: A negative boundary

Solve and graph x + 7 > 4.

  1. Subtract 7 from both sides: x > −3.
  2. Test x = 0: 0 + 7 = 7 > 4 ✓. Test x = −4: 3 > 4 ✗.
  3. Graph: open circle at −3 (−3 is not included), shaded to the right.

x > −3

x > −3−6−5−4−3−2−101234
Open circle at −3, shaded to the right.
Worked example

Example 2: A positive coefficient

Solve and graph 2x − 5 ≤ −1.

  1. Add 5 to both sides: 2x ≤ 4.
  2. Divide both sides by 2 (positive, so no reversing): x ≤ 2.
  3. Test x = 0: −5 ≤ −1 ✓. Test x = 3: 1 ≤ −1 ✗.

x ≤ 2

x ≤ 2−4−3−2−10123456
Closed circle at 2, shaded to the left.
Worked example

Example 3: Dividing by a negative

Solve and graph −3n + 4 ≥ 16.

  1. Subtract 4 from both sides: −3n ≥ 12.
  2. Divide both sides by −3 and reverse the sign: n ≤ −4.
  3. Test n = −5: 15 + 4 = 19 ≥ 16 ✓. Test n = 0: 4 ≥ 16 ✗. The reversed sign was right.

n ≤ −4

n ≤ −4−8−7−6−5−4−3−2−1012
Closed circle at −4, shaded to the left.
Worked example

Example 4: Multiple terms

Solve 5x − 8x + 2 < 14.

  1. Combine like terms: −3x + 2 < 14.
  2. Subtract 2: −3x < 12.
  3. Divide by −3 and reverse: x > −4.
  4. Test x = 0: 2 < 14 ✓. Test x = −5: −25 + 40 + 2 = 17 < 14 ✗.

x > −4

x > −4−8−7−6−5−4−3−2−1012
Open circle at −4, shaded to the right.

Another way: avoid the negative

For −3n + 4 ≥ 16, you can add 3n to both sides instead: 4 ≥ 16 + 3n. Then −12 ≥ 3n, so −4 ≥ n. That says the same thing as n ≤ −4. Either way, always test a value to check the direction.

Hands-on activity

A number line on the floor

You need: 21 sheets of paper (or sticky notes), a marker, and a partner.

  1. Write the integers from −10 to 10 on the sheets and lay them in a line on the floor.
  2. Flip test: you stand on 2 and your partner stands on 5. Who is on the bigger number? Now both multiply your number by −1 and move there (−2 and −5). Who is on the bigger number now? The order reversed!
  3. Solve by standing: for x − 3 ≥ −5, stand on each number and check whether it works. Lay a coin on every integer that works. Where do the coins start? (At −2, and they go to the right: x ≥ −2.)
  4. Solve −2x > 6 on paper, then check by standing on −4, −3 and −2.

What to look for: −2x > 6 gives x < −3. −4 works (8 > 6), −3 does not (6 > 6 is false), and −2 does not.

Just for fun

An inequality sign is like a hungry crocodile: its open mouth always faces the bigger number. Divide both sides by a negative, and the crocodile has to turn around, because the bigger number just switched sides.

Your turn

1

Solve and describe the graph of x − 4 > −6.

2

Solve 3y ≤ −12.

3

Solve −2a < 8.

4

Solve 4n + 9 ≥ 1.

5

Solve −5k − 3 > 12.

6

Is −7 a solution of 2x + 10 < −3?

7

A graph has a closed circle at −1, shaded to the left. Write the inequality.

More practice: warm-up

1

Solve x + 3 < 1.

2

Solve x − 5 ≥ −5.

3

Is −2 a solution of x > −3?

4

Solve 2x > −10.

5

Do you use an open or closed circle for x ≤ 6?

More practice: on your own

1

Solve and check −4x + 7 ≤ 19.

2

Solve 6m − 9m + 5 > −10.

3

Solve 2x + 3 ≥ 5x − 9.

4

In a game, Ava's score is −15. She earns 4 points for each correct answer. How many correct answers does she need for a score greater than 10? Graph the solutions.

5

"The temperature t stayed above −8°C all day." Write the inequality and describe its graph.

Challenge

1

Solve −2(x − 3) < 10.

2

List all the integers that make both x > −4 and x ≤ 1 true.

3

Use an example to explain why you reverse the sign when you divide both sides of an inequality by a negative number.

Check your understanding

Quiz

1.

Solve x + 5 > 2.

2.

Solve −2x ≤ 8.

3.

Which graph shows x < −1?

4.

Which number is a solution of 3x − 2 > 4?

5.

When must you reverse the inequality sign?

6.

Solve −3n + 1 < −8.

For parents and teachers

Big idea: C2.4 asks Grade 8 students to solve inequalities that involve integers, and to verify and graph the solutions. The key new idea is that multiplying or dividing both sides by a negative reverses the inequality sign.

Common difficulties:

  • Reversing the sign at the wrong time (for example when subtracting a negative, or dividing by a positive). Only multiplying or dividing by a negative reverses it.
  • Open versus closed circles. < and > use open circles; ≤ and ≥ use closed circles.
  • Shading the wrong way when the variable is on the right (−4 ≥ n). Rewrite it as n ≤ −4.
  • Skipping the test. Testing a value is the best way to catch a direction mistake.

How to help: Do the floor number line together. Physically moving to −2 and −5 makes the "order flips" idea memorable. This lesson builds A1 skills: reasoning, proving and checking.