GRAPHING INEQUALITIES
Exploration
One way to solve the double inequality −4 < 3𝑥 + 2 < 5 is:
Add −2 to both sides: −6 < 3𝑥 < 3
1
Multiply both sides by 3: −2 < 𝑥 < 1
Another way is to rewrite the double inequality as two distinct inequalities:
−4 < 3𝑥 + 2 and 3𝑥 + 2 < 5.
Solve them separately:
−6 < 3𝑥 and 3𝑥 < 3
−2 < 𝑥 and 𝑥 < 1
Combine the solution sets: −2 < 𝑥 < 1
1. Solve the double inequality −9 < 5 − 7𝑥 ≤ 12 in two ways algebraically.
2. Solve −14 < −7(3𝑥 + 2) < 21
a. By first using equivalence transformations
b. By expanding the brackets first.
Explain which method you prefer.
3. Does the pair of inequalities 4𝑥 < 4 and 3𝑥 − 5 ≥ 1 have a common solution set? Explain.
4. Given that −1 < 𝑥 < 4, find two values 𝑚 and 𝑛 such that 𝑚 < 2𝑥 + 3 < 𝑛.
SOLVING GRAPHICALLY
Let us solve the inequality 3𝑥 − 7 < 5.
Draw the lines 𝑓1 (𝑥) = 3𝑥 − 7 and 𝑓2 (𝑥) = 5 and find the intersection point. The symbol < means
that 3𝑥 − 7 cannot take the value 5. Use a dashed line for
𝑓2 (𝑥) = 5 to show that points on the line 𝑓2 (𝑥) = 5 are not
possible solutions.
The lines intersect at 𝑥 = 4. This is the value of 𝑥 that makes
both sides equal.
For 3𝑥 − 7 < 5 you need values of 𝑥 such that 𝑓1 (𝑥) < 𝑓2 (𝑥).
From the graph, the line for 𝑓1 (𝑥) is below 𝑓2 (𝑥) when 𝑥 < 4,
showing that the values of 𝑓1 (𝑥) are less than the values of
𝑓2 (𝑥) when 𝑥 < 4.
The solution is 𝑥 < 4.
You can check your solution by choosing a point with 𝑥 < 4,
for example 𝑥 = 2. Substitute 𝑥 = 2 into the inequality:
3𝑥 − 7 ⟶ 3 × 2 − 7 = 6 − 7 = −1 < 5 
Now choose a point with 𝑥 > 4, say: 𝑥 = 6.
Substitute 𝑥 = 6 into the inequality:
3𝑥 − 7 ⟶ 3 × 6 − 7 = 18 − 7 = 11. 11 is not less than 5, so the point does not satisfy the
inequality.
CHALLENGE
Solve the inequality 5𝑥 − 7 ≥ 3𝑥 + 9 graphically and then check your answer algebraically.
Practice 2
Solve the following inequalities graphically:
1. 3𝑥 − 7 < 5
𝑥
2. 6 − 7 > 2
4. 5𝑥 − 7 ≥ 3𝑥 + 9
3
5. 1 − 𝑥 ≤ 𝑥 − 4
3. 3(4 − 𝑚) > 9
6.
2
2𝑥+1
−
3
> 5𝑥 + 2
7. −2(𝑎 − 3) < 5(𝑎 − 2) − 12
8. 2(1 − 𝑏) + 5 ≥ 3(2𝑏 − 1)
9. 4𝑘 − 11 ≤
3𝑘
2
+5
Practice 3
Write down the two inequalities that could be represented by the non-shaded region of this graph.
Practice 4
Verify, graphically and algebraically, that the inequality 3𝑥 + 2 < 9𝑥 + 6 has an infinite number of
solutions.
Which of the methods do you prefer? Explain why.