Solving Quadratic Inequalities Textbook Exercise - Corbettmaths - Free Printable

Let's solve the quadratic inequalities step by step. We'll go through Question 1 and Question 2, explaining the method for solving each type of inequality.

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🔍 Method for Solving Quadratic Inequalities:

1. Factorize the quadratic expression (if possible).
2. Find the critical points (roots) by setting the expression equal to zero.
3. Use a number line or sign chart to test intervals between the roots.
4. Determine where the inequality holds true based on the sign of the expression in each interval.
5. Include endpoints if the inequality is "≤" or "≥".

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## ✔ Question 1: Solve the following inequalities

We are given factored forms, so we can directly find the roots and analyze signs.

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(a) $(x - 4)(x - 1) < 0$

- Roots: $x = 1$, $x = 4$
- The product is negative when one factor is positive and the other is negative.
- Sign chart:
- $x < 1$: both negative → positive
- $1 < x < 4$: $x - 1 > 0$, $x - 4 < 0$ → negative ✔
- $x > 4$: both positive → positive
- So, solution: $1 < x < 4$

✔ Answer: $\boxed{(1, 4)}$

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(b) $(x - 2)(x + 1) < 0$

- Roots: $x = -1$, $x = 2$
- Product negative between roots.
- $-1 < x < 2$

✔ Answer: $\boxed{(-1, 2)}$

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(c) $(x + 7)(x + 3) \leq 0$

- Roots: $x = -7$, $x = -3$
- Product ≤ 0 → includes equality at roots.
- Negative between roots: $-7 < x < -3$
- Include endpoints since ≤

✔ Answer: $\boxed{[-7, -3]}$

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(d) $(x - 5)(x + 4) \leq 0$

- Roots: $x = -4$, $x = 5$
- Product ≤ 0 → between roots, including endpoints
- $-4 \leq x \leq 5$

✔ Answer: $\boxed{[-4, 5]}$

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(e) $x(x - 9) > 0$

- Roots: $x = 0$, $x = 9$
- Product > 0 → outside the roots
- $x < 0$: both negative → positive ✔
- $x > 9$: both positive → positive ✔
- $0 < x < 9$: negative × positive = negative ✘

✔ Answer: $\boxed{(-\infty, 0) \cup (9, \infty)}$

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(f) $(x + 6)(x - 5) > 0$

- Roots: $x = -6$, $x = 5$
- Positive outside roots
- $x < -6$: both negative → positive ✔
- $x > 5$: both positive → positive ✔
- $-6 < x < 5$: mixed → negative ✘

✔ Answer: $\boxed{(-\infty, -6) \cup (5, \infty)}$

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(g) $(x + 10)(x + 1) \geq 0$

- Roots: $x = -10$, $x = -1$
- ≥ 0 → outside or at roots
- $x \leq -10$ or $x \geq -1$

✔ Answer: $\boxed{(-\infty, -10] \cup [-1, \infty)}$

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(h) $(x - 7)(x + 7) \geq 0$

- Roots: $x = -7$, $x = 7$
- ≥ 0 → outside or at roots
- $x \leq -7$ or $x \geq 7$

✔ Answer: $\boxed{(-\infty, -7] \cup [7, \infty)}$

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(i) $(x + 8)(x + 2) < 0$

- Roots: $x = -8$, $x = -2$
- Negative between roots
- $-8 < x < -2$

✔ Answer: $\boxed{(-8, -2)}$

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(j) $(x - 4)(x + 7) \geq 0$

- Roots: $x = -7$, $x = 4$
- ≥ 0 → outside or at roots
- $x \leq -7$ or $x \geq 4$

✔ Answer: $\boxed{(-\infty, -7] \cup [4, \infty)}$

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(k) $(x + 1)(x - 5) \leq 0$

- Roots: $x = -1$, $x = 5$
- ≤ 0 → between roots, inclusive
- $-1 \leq x \leq 5$

✔ Answer: $\boxed{[-1, 5]}$

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(l) $(x - 12)(x - 11) > 0$

- Roots: $x = 11$, $x = 12$
- > 0 → outside roots
- $x < 11$ or $x > 12$

✔ Answer: $\boxed{(-\infty, 11) \cup (12, \infty)}$

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## ✔ Question 2: Solve the following inequalities

Now we need to factor first.

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(a) $x^2 + 5x + 6 > 0$

- Factor: $(x + 2)(x + 3) > 0$
- Roots: $x = -3$, $x = -2$
- > 0 → outside roots
- $x < -3$ or $x > -2$

✔ Answer: $\boxed{(-\infty, -3) \cup (-2, \infty)}$

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(b) $x^2 + 7x + 10 < 0$

- Factor: $(x + 2)(x + 5) < 0$
- Roots: $x = -5$, $x = -2$
- < 0 → between roots
- $-5 < x < -2$

✔ Answer: $\boxed{(-5, -2)}$

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(c) $x^2 - 4x - 5 \leq 0$

- Factor: $(x - 5)(x + 1) \leq 0$
- Roots: $x = -1$, $x = 5$
- ≤ 0 → between roots, inclusive
- $-1 \leq x \leq 5$

✔ Answer: $\boxed{[-1, 5]}$

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(d) $x^2 + 2x - 24 > 0$

- Factor: $(x + 6)(x - 4) > 0$
- Roots: $x = -6$, $x = 4$
- > 0 → outside roots
- $x < -6$ or $x > 4$

✔ Answer: $\boxed{(-\infty, -6) \cup (4, \infty)}$

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(e) $x^2 - 6x + 8 \geq 0$

- Factor: $(x - 2)(x - 4) \geq 0$
- Roots: $x = 2$, $x = 4$
- ≥ 0 → outside or at roots
- $x \leq 2$ or $x \geq 4$

✔ Answer: $\boxed{(-\infty, 2] \cup [4, \infty)}$

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(f) $x^2 + 3x - 4 < 0$

- Factor: $(x + 4)(x - 1) < 0$
- Roots: $x = -4$, $x = 1$
- < 0 → between roots
- $-4 < x < 1$

✔ Answer: $\boxed{(-4, 1)}$

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(g) $x^2 - x - 56 > 0$

- Factor: Find two numbers that multiply to $-56$, add to $-1$: $-8$ and $7$
- $(x - 8)(x + 7) > 0$
- Roots: $x = -7$, $x = 8$
- > 0 → outside roots
- $x < -7$ or $x > 8$

✔ Answer: $\boxed{(-\infty, -7) \cup (8, \infty)}$

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(h) $x^2 + 9x + 18 < 0$

- Factor: $(x + 3)(x + 6) < 0$
- Roots: $x = -6$, $x = -3$
- < 0 → between roots
- $-6 < x < -3$

✔ Answer: $\boxed{(-6, -3)}$

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(i) $x^2 - 13x + 22 \leq 0$

- Factor: Find two numbers: $-11$ and $-2$ → $(x - 11)(x - 2) \leq 0$
- Roots: $x = 2$, $x = 11$
- ≤ 0 → between roots, inclusive
- $2 \leq x \leq 11$

✔ Answer: $\boxed{[2, 11]}$

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(j) $x^2 - 4x - 32 < 0$

- Factor: Numbers: $-8$ and $4$ → $(x - 8)(x + 4) < 0$
- Roots: $x = -4$, $x = 8$
- < 0 → between roots
- $-4 < x < 8$

✔ Answer: $\boxed{(-4, 8)}$

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(k) $x^2 - 64 \geq 0$

- This is difference of squares: $(x - 8)(x + 8) \geq 0$
- Roots: $x = -8$, $x = 8$
- ≥ 0 → outside or at roots
- $x \leq -8$ or $x \geq 8$

✔ Answer: $\boxed{(-\infty, -8] \cup [8, \infty)}$

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(l) $x^2 - 14x + 48 > 0$

- Factor: $-6$ and $-8$ → $(x - 6)(x - 8) > 0$
- Roots: $x = 6$, $x = 8$
- > 0 → outside roots
- $x < 6$ or $x > 8$

✔ Answer: $\boxed{(-\infty, 6) \cup (8, \infty)}$

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## 📌 Summary Table

| Question | Answer |
|--------|--------|
| (a) | $(1, 4)$ |
| (b) | $(-1, 2)$ |
| (c) | $[-7, -3]$ |
| (d) | $[-4, 5]$ |
| (e) | $(-\infty, 0) \cup (9, \infty)$ |
| (f) | $(-\infty, -6) \cup (5, \infty)$ |
| (g) | $(-\infty, -10] \cup [-1, \infty)$ |
| (h) | $(-\infty, -7] \cup [7, \infty)$ |
| (i) | $(-8, -2)$ |
| (j) | $(-\infty, -7] \cup [4, \infty)$ |
| (k) | $[-1, 5]$ |
| (l) | $(-\infty, 11) \cup (12, \infty)$ |

| Question | Answer |
|--------|--------|
| (a) | $(-\infty, -3) \cup (-2, \infty)$ |
| (b) | $(-5, -2)$ |
| (c) | $[-1, 5]$ |
| (d) | $(-\infty, -6) \cup (4, \infty)$ |
| (e) | $(-\infty, 2] \cup [4, \infty)$ |
| (f) | $(-4, 1)$ |
| (g) | $(-\infty, -7) \cup (8, \infty)$ |
| (h) | $(-6, -3)$ |
| (i) | $[2, 11]$ |
| (j) | $(-4, 8)$ |
| (k) | $(-\infty, -8] \cup [8, \infty)$ |
| (l) | $(-\infty, 6) \cup (8, \infty)$ |

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✔ Final Note: Always check the inequality sign and whether equality is allowed (use brackets accordingly). Use a sign chart or number line to verify your solution!

Let me know if you'd like a visual explanation or graph for any of these!