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Step-by-step solution for: Solving Quadratic Inequalities Textbook Exercise - Corbettmaths
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Step-by-step solution for: Solving Quadratic Inequalities Textbook Exercise - Corbettmaths
Let’s solve each inequality step by step. We’ll start with Question 1, then move to Question 2.
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Question 1: Solve the following inequalities
These are all factored quadratic inequalities. The key idea is:
- Find the “critical points” (where each factor equals zero).
- These points divide the number line into regions.
- Test a value in each region to see if it makes the inequality true.
- Remember: for < or >, we don’t include the endpoints; for ≤ or ≥, we do.
---
(a) (x – 4)(x – 1) < 0
Critical points: x = 1 and x = 4
Test intervals:
- x < 1 → try x=0: (-4)(-1)=4 > 0 → not good
- 1 < x < 4 → try x=2: (-2)(1)=-2 < 0 → GOOD
- x > 4 → try x=5: (1)(4)=4 > 0 → not good
Answer: 1 < x < 4
---
(b) (x – 2)(x + 1) < 0
Critical points: x = -1 and x = 2
Test intervals:
- x < -1 → try x=-2: (-4)(-1)=4 > 0 → no
- -1 < x < 2 → try x=0: (-2)(1)=-2 < 0 → YES
- x > 2 → try x=3: (1)(4)=4 > 0 → no
Answer: -1 < x < 2
---
(c) (x + 7)(x + 3) ≤ 0
Critical points: x = -7 and x = -3
Since it’s ≤, we include endpoints.
Test intervals:
- x < -7 → try x=-8: (-1)(-5)=5 > 0 → no
- -7 ≤ x ≤ -3 → try x=-5: (2)(-2)=-4 ≤ 0 → YES
- x > -3 → try x=0: (7)(3)=21 > 0 → no
Answer: -7 ≤ x ≤ -3
---
(d) (x – 5)(x + 4) ≤ 0
Critical points: x = -4 and x = 5
Include endpoints.
Test:
- x < -4 → try x=-5: (-10)(-1)=10 > 0 → no
- -4 ≤ x ≤ 5 → try x=0: (-5)(4)=-20 ≤ 0 → YES
- x > 5 → try x=6: (1)(10)=10 > 0 → no
Answer: -4 ≤ x ≤ 5
---
(e) x(x – 9) > 0
Critical points: x = 0 and x = 9
Do NOT include endpoints (since >).
Test:
- x < 0 → try x=-1: (-1)(-10)=10 > 0 → YES
- 0 < x < 9 → try x=1: (1)(-8)=-8 < 0 → no
- x > 9 → try x=10: (10)(1)=10 > 0 → YES
Answer: x < 0 or x > 9
---
(f) (x + 6)(x – 5) > 0
Critical points: x = -6 and x = 5
Test:
- x < -6 → try x=-7: (-1)(-12)=12 > 0 → YES
- -6 < x < 5 → try x=0: (6)(-5)=-30 < 0 → no
- x > 5 → try x=6: (12)(1)=12 > 0 → YES
Answer: x < -6 or x > 5
---
(g) (x + 10)(x + 1) ≥ 0
Critical points: x = -10 and x = -1
Include endpoints.
Test:
- x < -10 → try x=-11: (-1)(-10)=10 ≥ 0 → YES
- -10 < x < -1 → try x=-5: (5)(-4)=-20 < 0 → no
- x > -1 → try x=0: (10)(1)=10 ≥ 0 → YES
Answer: x ≤ -10 or x ≥ -1
---
(h) (x – 7)(x + 7) ≥ 0
This is same as (x² - 49) ≥ 0
Critical points: x = -7 and x = 7
Test:
- x < -7 → try x=-8: (-15)(-1)=15 ≥ 0 → YES
- -7 < x < 7 → try x=0: (-7)(7)=-49 < 0 → no
- x > 7 → try x=8: (1)(15)=15 ≥ 0 → YES
Answer: x ≤ -7 or x ≥ 7
---
(i) (x + 8)(x + 2) < 0
Critical points: x = -8 and x = -2
Test:
- x < -8 → try x=-9: (-1)(-7)=7 > 0 → no
- -8 < x < -2 → try x=-5: (3)(-3)=-9 < 0 → YES
- x > -2 → try x=0: (8)(2)=16 > 0 → no
Answer: -8 < x < -2
---
(j) (x – 4)(x + 7) ≥ 0
Critical points: x = -7 and x = 4
Test:
- x < -7 → try x=-8: (-12)(-1)=12 ≥ 0 → YES
- -7 < x < 4 → try x=0: (-4)(7)=-28 < 0 → no
- x > 4 → try x=5: (1)(12)=12 ≥ 0 → YES
Answer: x ≤ -7 or x ≥ 4
---
(k) (x + 1)(x – 5) ≤ 0
Critical points: x = -1 and x = 5
Test:
- x < -1 → try x=-2: (-1)(-7)=7 > 0 → no
- -1 ≤ x ≤ 5 → try x=0: (1)(-5)=-5 ≤ 0 → YES
- x > 5 → try x=6: (7)(1)=7 > 0 → no
Answer: -1 ≤ x ≤ 5
---
(l) (x – 12)(x – 11) > 0
Critical points: x = 11 and x = 12
Test:
- x < 11 → try x=10: (-2)(-1)=2 > 0 → YES
- 11 < x < 12 → try x=11.5: (-0.5)(0.5)=-0.25 < 0 → no
- x > 12 → try x=13: (1)(2)=2 > 0 → YES
Answer: x < 11 or x > 12
---
Now let’s go to Question 2: These are unfactored quadratics. First, factor them (if possible), then solve like above.
---
(a) x² + 5x + 6 > 0
Factor: (x+2)(x+3) > 0
Critical points: x = -3, x = -2
Test:
- x < -3 → try x=-4: (-2)(-1)=2 > 0 → YES
- -3 < x < -2 → try x=-2.5: (-0.5)(0.5)=-0.25 < 0 → no
- x > -2 → try x=0: (2)(3)=6 > 0 → YES
Answer: x < -3 or x > -2
---
(b) x² + 7x + 10 < 0
Factor: (x+2)(x+5) < 0
Critical points: x = -5, x = -2
Test:
- x < -5 → try x=-6: (-4)(-1)=4 > 0 → no
- -5 < x < -2 → try x=-3: (-1)(2)=-2 < 0 → YES
- x > -2 → try x=0: (2)(5)=10 > 0 → no
Answer: -5 < x < -2
---
(c) x² – 4x – 5 ≤ 0
Factor: (x–5)(x+1) ≤ 0
Critical points: x = -1, x = 5
Test:
- x < -1 → try x=-2: (-7)(-1)=7 > 0 → no
- -1 ≤ x ≤ 5 → try x=0: (-5)(1)=-5 ≤ 0 → YES
- x > 5 → try x=6: (1)(7)=7 > 0 → no
Answer: -1 ≤ x ≤ 5
---
(d) x² + 2x – 24 > 0
Factor: (x+6)(x–4) > 0
Critical points: x = -6, x = 4
Test:
- x < -6 → try x=-7: (-1)(-11)=11 > 0 → YES
- -6 < x < 4 → try x=0: (6)(-4)=-24 < 0 → no
- x > 4 → try x=5: (11)(1)=11 > 0 → YES
Answer: x < -6 or x > 4
---
(e) x² – 6x + 8 ≥ 0
Factor: (x–2)(x–4) ≥ 0
Critical points: x = 2, x = 4
Test:
- x < 2 → try x=0: (-2)(-4)=8 ≥ 0 → YES
- 2 < x < 4 → try x=3: (1)(-1)=-1 < 0 → no
- x > 4 → try x=5: (3)(1)=3 ≥ 0 → YES
Answer: x ≤ 2 or x ≥ 4
---
(f) x² + 3x – 4 < 0
Factor: (x+4)(x–1) < 0
Critical points: x = -4, x = 1
Test:
- x < -4 → try x=-5: (-1)(-6)=6 > 0 → no
- -4 < x < 1 → try x=0: (4)(-1)=-4 < 0 → YES
- x > 1 → try x=2: (6)(1)=6 > 0 → no
Answer: -4 < x < 1
---
(g) x² – x – 56 > 0
Factor: (x–8)(x+7) > 0
Critical points: x = -7, x = 8
Test:
- x < -7 → try x=-8: (-16)(-1)=16 > 0 → YES
- -7 < x < 8 → try x=0: (-8)(7)=-56 < 0 → no
- x > 8 → try x=9: (1)(16)=16 > 0 → YES
Answer: x < -7 or x > 8
---
(h) x² + 9x + 18 < 0
Factor: (x+3)(x+6) < 0
Critical points: x = -6, x = -3
Test:
- x < -6 → try x=-7: (-4)(-1)=4 > 0 → no
- -6 < x < -3 → try x=-4: (-1)(2)=-2 < 0 → YES
- x > -3 → try x=0: (3)(6)=18 > 0 → no
Answer: -6 < x < -3
---
(i) x² – 13x + 22 ≤ 0
Factor: (x–11)(x–2) ≤ 0
Critical points: x = 2, x = 11
Test:
- x < 2 → try x=0: (-11)(-2)=22 > 0 → no
- 2 ≤ x ≤ 11 → try x=5: (-6)(3)=-18 ≤ 0 → YES
- x > 11 → try x=12: (1)(10)=10 > 0 → no
Answer: 2 ≤ x ≤ 11
---
(j) x² – 4x – 32 < 0
Factor: (x–8)(x+4) < 0
Critical points: x = -4, x = 8
Test:
- x < -4 → try x=-5: (-13)(-1)=13 > 0 → no
- -4 < x < 8 → try x=0: (-8)(4)=-32 < 0 → YES
- x > 8 → try x=9: (1)(13)=13 > 0 → no
Answer: -4 < x < 8
---
(k) x² – 64 ≥ 0
This is difference of squares: (x–8)(x+8) ≥ 0
Critical points: x = -8, x = 8
Test:
- x < -8 → try x=-9: (-17)(-1)=17 ≥ 0 → YES
- -8 < x < 8 → try x=0: (-8)(8)=-64 < 0 → no
- x > 8 → try x=9: (1)(17)=17 ≥ 0 → YES
Answer: x ≤ -8 or x ≥ 8
---
(l) x² – 14x + 48 > 0
Factor: (x–6)(x–8) > 0
Critical points: x = 6, x = 8
Test:
- x < 6 → try x=0: (-6)(-8)=48 > 0 → YES
- 6 < x < 8 → try x=7: (1)(-1)=-1 < 0 → no
- x > 8 → try x=9: (3)(1)=3 > 0 → YES
Answer: x < 6 or x > 8
---
Final Answer:
Question 1:
(a) 1 < x < 4
(b) -1 < x < 2
(c) -7 ≤ x ≤ -3
(d) -4 ≤ x ≤ 5
(e) x < 0 or x > 9
(f) x < -6 or x > 5
(g) x ≤ -10 or x ≥ -1
(h) x ≤ -7 or x ≥ 7
(i) -8 < x < -2
(j) x ≤ -7 or x ≥ 4
(k) -1 ≤ x ≤ 5
(l) x < 11 or x > 12
Question 2:
(a) x < -3 or x > -2
(b) -5 < x < -2
(c) -1 ≤ x ≤ 5
(d) x < -6 or x > 4
(e) x ≤ 2 or x ≥ 4
(f) -4 < x < 1
(g) x < -7 or x > 8
(h) -6 < x < -3
(i) 2 ≤ x ≤ 11
(j) -4 < x < 8
(k) x ≤ -8 or x ≥ 8
(l) x < 6 or x > 8
---
Question 1: Solve the following inequalities
These are all factored quadratic inequalities. The key idea is:
- Find the “critical points” (where each factor equals zero).
- These points divide the number line into regions.
- Test a value in each region to see if it makes the inequality true.
- Remember: for < or >, we don’t include the endpoints; for ≤ or ≥, we do.
---
(a) (x – 4)(x – 1) < 0
Critical points: x = 1 and x = 4
Test intervals:
- x < 1 → try x=0: (-4)(-1)=4 > 0 → not good
- 1 < x < 4 → try x=2: (-2)(1)=-2 < 0 → GOOD
- x > 4 → try x=5: (1)(4)=4 > 0 → not good
Answer: 1 < x < 4
---
(b) (x – 2)(x + 1) < 0
Critical points: x = -1 and x = 2
Test intervals:
- x < -1 → try x=-2: (-4)(-1)=4 > 0 → no
- -1 < x < 2 → try x=0: (-2)(1)=-2 < 0 → YES
- x > 2 → try x=3: (1)(4)=4 > 0 → no
Answer: -1 < x < 2
---
(c) (x + 7)(x + 3) ≤ 0
Critical points: x = -7 and x = -3
Since it’s ≤, we include endpoints.
Test intervals:
- x < -7 → try x=-8: (-1)(-5)=5 > 0 → no
- -7 ≤ x ≤ -3 → try x=-5: (2)(-2)=-4 ≤ 0 → YES
- x > -3 → try x=0: (7)(3)=21 > 0 → no
Answer: -7 ≤ x ≤ -3
---
(d) (x – 5)(x + 4) ≤ 0
Critical points: x = -4 and x = 5
Include endpoints.
Test:
- x < -4 → try x=-5: (-10)(-1)=10 > 0 → no
- -4 ≤ x ≤ 5 → try x=0: (-5)(4)=-20 ≤ 0 → YES
- x > 5 → try x=6: (1)(10)=10 > 0 → no
Answer: -4 ≤ x ≤ 5
---
(e) x(x – 9) > 0
Critical points: x = 0 and x = 9
Do NOT include endpoints (since >).
Test:
- x < 0 → try x=-1: (-1)(-10)=10 > 0 → YES
- 0 < x < 9 → try x=1: (1)(-8)=-8 < 0 → no
- x > 9 → try x=10: (10)(1)=10 > 0 → YES
Answer: x < 0 or x > 9
---
(f) (x + 6)(x – 5) > 0
Critical points: x = -6 and x = 5
Test:
- x < -6 → try x=-7: (-1)(-12)=12 > 0 → YES
- -6 < x < 5 → try x=0: (6)(-5)=-30 < 0 → no
- x > 5 → try x=6: (12)(1)=12 > 0 → YES
Answer: x < -6 or x > 5
---
(g) (x + 10)(x + 1) ≥ 0
Critical points: x = -10 and x = -1
Include endpoints.
Test:
- x < -10 → try x=-11: (-1)(-10)=10 ≥ 0 → YES
- -10 < x < -1 → try x=-5: (5)(-4)=-20 < 0 → no
- x > -1 → try x=0: (10)(1)=10 ≥ 0 → YES
Answer: x ≤ -10 or x ≥ -1
---
(h) (x – 7)(x + 7) ≥ 0
This is same as (x² - 49) ≥ 0
Critical points: x = -7 and x = 7
Test:
- x < -7 → try x=-8: (-15)(-1)=15 ≥ 0 → YES
- -7 < x < 7 → try x=0: (-7)(7)=-49 < 0 → no
- x > 7 → try x=8: (1)(15)=15 ≥ 0 → YES
Answer: x ≤ -7 or x ≥ 7
---
(i) (x + 8)(x + 2) < 0
Critical points: x = -8 and x = -2
Test:
- x < -8 → try x=-9: (-1)(-7)=7 > 0 → no
- -8 < x < -2 → try x=-5: (3)(-3)=-9 < 0 → YES
- x > -2 → try x=0: (8)(2)=16 > 0 → no
Answer: -8 < x < -2
---
(j) (x – 4)(x + 7) ≥ 0
Critical points: x = -7 and x = 4
Test:
- x < -7 → try x=-8: (-12)(-1)=12 ≥ 0 → YES
- -7 < x < 4 → try x=0: (-4)(7)=-28 < 0 → no
- x > 4 → try x=5: (1)(12)=12 ≥ 0 → YES
Answer: x ≤ -7 or x ≥ 4
---
(k) (x + 1)(x – 5) ≤ 0
Critical points: x = -1 and x = 5
Test:
- x < -1 → try x=-2: (-1)(-7)=7 > 0 → no
- -1 ≤ x ≤ 5 → try x=0: (1)(-5)=-5 ≤ 0 → YES
- x > 5 → try x=6: (7)(1)=7 > 0 → no
Answer: -1 ≤ x ≤ 5
---
(l) (x – 12)(x – 11) > 0
Critical points: x = 11 and x = 12
Test:
- x < 11 → try x=10: (-2)(-1)=2 > 0 → YES
- 11 < x < 12 → try x=11.5: (-0.5)(0.5)=-0.25 < 0 → no
- x > 12 → try x=13: (1)(2)=2 > 0 → YES
Answer: x < 11 or x > 12
---
Now let’s go to Question 2: These are unfactored quadratics. First, factor them (if possible), then solve like above.
---
(a) x² + 5x + 6 > 0
Factor: (x+2)(x+3) > 0
Critical points: x = -3, x = -2
Test:
- x < -3 → try x=-4: (-2)(-1)=2 > 0 → YES
- -3 < x < -2 → try x=-2.5: (-0.5)(0.5)=-0.25 < 0 → no
- x > -2 → try x=0: (2)(3)=6 > 0 → YES
Answer: x < -3 or x > -2
---
(b) x² + 7x + 10 < 0
Factor: (x+2)(x+5) < 0
Critical points: x = -5, x = -2
Test:
- x < -5 → try x=-6: (-4)(-1)=4 > 0 → no
- -5 < x < -2 → try x=-3: (-1)(2)=-2 < 0 → YES
- x > -2 → try x=0: (2)(5)=10 > 0 → no
Answer: -5 < x < -2
---
(c) x² – 4x – 5 ≤ 0
Factor: (x–5)(x+1) ≤ 0
Critical points: x = -1, x = 5
Test:
- x < -1 → try x=-2: (-7)(-1)=7 > 0 → no
- -1 ≤ x ≤ 5 → try x=0: (-5)(1)=-5 ≤ 0 → YES
- x > 5 → try x=6: (1)(7)=7 > 0 → no
Answer: -1 ≤ x ≤ 5
---
(d) x² + 2x – 24 > 0
Factor: (x+6)(x–4) > 0
Critical points: x = -6, x = 4
Test:
- x < -6 → try x=-7: (-1)(-11)=11 > 0 → YES
- -6 < x < 4 → try x=0: (6)(-4)=-24 < 0 → no
- x > 4 → try x=5: (11)(1)=11 > 0 → YES
Answer: x < -6 or x > 4
---
(e) x² – 6x + 8 ≥ 0
Factor: (x–2)(x–4) ≥ 0
Critical points: x = 2, x = 4
Test:
- x < 2 → try x=0: (-2)(-4)=8 ≥ 0 → YES
- 2 < x < 4 → try x=3: (1)(-1)=-1 < 0 → no
- x > 4 → try x=5: (3)(1)=3 ≥ 0 → YES
Answer: x ≤ 2 or x ≥ 4
---
(f) x² + 3x – 4 < 0
Factor: (x+4)(x–1) < 0
Critical points: x = -4, x = 1
Test:
- x < -4 → try x=-5: (-1)(-6)=6 > 0 → no
- -4 < x < 1 → try x=0: (4)(-1)=-4 < 0 → YES
- x > 1 → try x=2: (6)(1)=6 > 0 → no
Answer: -4 < x < 1
---
(g) x² – x – 56 > 0
Factor: (x–8)(x+7) > 0
Critical points: x = -7, x = 8
Test:
- x < -7 → try x=-8: (-16)(-1)=16 > 0 → YES
- -7 < x < 8 → try x=0: (-8)(7)=-56 < 0 → no
- x > 8 → try x=9: (1)(16)=16 > 0 → YES
Answer: x < -7 or x > 8
---
(h) x² + 9x + 18 < 0
Factor: (x+3)(x+6) < 0
Critical points: x = -6, x = -3
Test:
- x < -6 → try x=-7: (-4)(-1)=4 > 0 → no
- -6 < x < -3 → try x=-4: (-1)(2)=-2 < 0 → YES
- x > -3 → try x=0: (3)(6)=18 > 0 → no
Answer: -6 < x < -3
---
(i) x² – 13x + 22 ≤ 0
Factor: (x–11)(x–2) ≤ 0
Critical points: x = 2, x = 11
Test:
- x < 2 → try x=0: (-11)(-2)=22 > 0 → no
- 2 ≤ x ≤ 11 → try x=5: (-6)(3)=-18 ≤ 0 → YES
- x > 11 → try x=12: (1)(10)=10 > 0 → no
Answer: 2 ≤ x ≤ 11
---
(j) x² – 4x – 32 < 0
Factor: (x–8)(x+4) < 0
Critical points: x = -4, x = 8
Test:
- x < -4 → try x=-5: (-13)(-1)=13 > 0 → no
- -4 < x < 8 → try x=0: (-8)(4)=-32 < 0 → YES
- x > 8 → try x=9: (1)(13)=13 > 0 → no
Answer: -4 < x < 8
---
(k) x² – 64 ≥ 0
This is difference of squares: (x–8)(x+8) ≥ 0
Critical points: x = -8, x = 8
Test:
- x < -8 → try x=-9: (-17)(-1)=17 ≥ 0 → YES
- -8 < x < 8 → try x=0: (-8)(8)=-64 < 0 → no
- x > 8 → try x=9: (1)(17)=17 ≥ 0 → YES
Answer: x ≤ -8 or x ≥ 8
---
(l) x² – 14x + 48 > 0
Factor: (x–6)(x–8) > 0
Critical points: x = 6, x = 8
Test:
- x < 6 → try x=0: (-6)(-8)=48 > 0 → YES
- 6 < x < 8 → try x=7: (1)(-1)=-1 < 0 → no
- x > 8 → try x=9: (3)(1)=3 > 0 → YES
Answer: x < 6 or x > 8
---
Final Answer:
Question 1:
(a) 1 < x < 4
(b) -1 < x < 2
(c) -7 ≤ x ≤ -3
(d) -4 ≤ x ≤ 5
(e) x < 0 or x > 9
(f) x < -6 or x > 5
(g) x ≤ -10 or x ≥ -1
(h) x ≤ -7 or x ≥ 7
(i) -8 < x < -2
(j) x ≤ -7 or x ≥ 4
(k) -1 ≤ x ≤ 5
(l) x < 11 or x > 12
Question 2:
(a) x < -3 or x > -2
(b) -5 < x < -2
(c) -1 ≤ x ≤ 5
(d) x < -6 or x > 4
(e) x ≤ 2 or x ≥ 4
(f) -4 < x < 1
(g) x < -7 or x > 8
(h) -6 < x < -3
(i) 2 ≤ x ≤ 11
(j) -4 < x < 8
(k) x ≤ -8 or x ≥ 8
(l) x < 6 or x > 8
Parent Tip: Review the logic above to help your child master the concept of solving quadratic inequalities worksheet.
![Printable quadratic inequalities worksheet [PDFs] Brighterly.com](/images2/solving-quadratic-inequalities-worksheet/solving-quadratic-inequalities-worksheet-22.jpg)
