Quadratic inequalities - Free Printable

Problem Analysis and Solution

The task involves solving a series of inequalities and finding solutions in interval notation. Let's solve each inequality step by step.

---

#### Row 1:

1. Inequality: \( x^2 - 5x + 6 \geq 0 \)
- Factorize: \( x^2 - 5x + 6 = (x - 2)(x - 3) \)
- Roots: \( x = 2 \) and \( x = 3 \)
- Sign analysis:
- For \( x < 2 \): Both factors are negative, so the product is positive.
- For \( 2 < x < 3 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 3 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-\infty, 2] \cup [3, +\infty) \)

2. Inequality: \( x^2 - x + 2 > 0 \)
- Discriminant: \( \Delta = b^2 - 4ac = (-1)^2 - 4(1)(2) = 1 - 8 = -7 \)
- Since the discriminant is negative, the quadratic has no real roots and is always positive.
- Solution: \( x \in (-\infty, +\infty) \)

3. Inequality: \( 3x \geq x^2 + 4 \)
- Rearrange: \( x^2 - 3x + 4 \leq 0 \)
- Discriminant: \( \Delta = b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7 \)
- Since the discriminant is negative, the quadratic has no real roots and is always positive.
- Solution: \( x \in \emptyset \)

---

#### Row 2:

4. Inequality: \( 2x - 1 \leq (x - 3)(x + 3) \)
- Expand the right-hand side: \( (x - 3)(x + 3) = x^2 - 9 \)
- Rearrange: \( 2x - 1 \leq x^2 - 9 \)
- Simplify: \( x^2 - 2x - 8 \geq 0 \)
- Factorize: \( x^2 - 2x - 8 = (x - 4)(x + 2) \)
- Roots: \( x = 4 \) and \( x = -2 \)
- Sign analysis:
- For \( x < -2 \): Both factors are negative, so the product is positive.
- For \( -2 < x < 4 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 4 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-\infty, -2] \cup [4, +\infty) \)

5. Inequality: \( x^2 + 1 \leq 0 \)
- The quadratic \( x^2 + 1 \) is always positive for all real \( x \) because \( x^2 \geq 0 \) and adding 1 makes it strictly positive.
- Solution: \( x \in \emptyset \)

6. Inequality: \( 3x^2 + 12 > 0 \)
- Factor out 3: \( 3(x^2 + 4) > 0 \)
- Since \( x^2 + 4 \) is always positive, the inequality holds for all real \( x \).
- Solution: \( x \in (-\infty, +\infty) \)

---

#### Row 3:

7. Inequality: \( \frac{x}{2} + \frac{x(x-1)}{6} \leq 4 \)
- Combine terms over a common denominator:
\[
\frac{3x + x(x-1)}{6} \leq 4 \implies \frac{3x + x^2 - x}{6} \leq 4 \implies \frac{x^2 + 2x}{6} \leq 4
\]
- Multiply through by 6: \( x^2 + 2x \leq 24 \)
- Rearrange: \( x^2 + 2x - 24 \leq 0 \)
- Factorize: \( x^2 + 2x - 24 = (x + 6)(x - 4) \)
- Roots: \( x = -6 \) and \( x = 4 \)
- Sign analysis:
- For \( x < -6 \): Both factors are negative, so the product is positive.
- For \( -6 < x < 4 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 4 \): Both factors are positive, so the product is positive.
- Solution: \( x \in [-6, 4] \)

8. Inequality: \( x^2 - 6x + 9 > 0 \)
- Factorize: \( x^2 - 6x + 9 = (x - 3)^2 \)
- The expression \( (x - 3)^2 \) is always non-negative and equals zero only at \( x = 3 \).
- Solution: \( x \in (-\infty, 3) \cup (3, +\infty) \)

9. Inequality: \( x^2 - 4x + 4 \leq 0 \)
- Factorize: \( x^2 - 4x + 4 = (x - 2)^2 \)
- The expression \( (x - 2)^2 \) is always non-negative and equals zero only at \( x = 2 \).
- Solution: \( x = 2 \)

---

#### Row 4:

10. Inequality: \( (x - 1)(x - 3) < 0 \)
- Roots: \( x = 1 \) and \( x = 3 \)
- Sign analysis:
- For \( x < 1 \): Both factors are negative, so the product is positive.
- For \( 1 < x < 3 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 3 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (1, 3) \)

11. Inequality: \( (x + 2)(x - 5) \geq 0 \)
- Roots: \( x = -2 \) and \( x = 5 \)
- Sign analysis:
- For \( x < -2 \): Both factors are negative, so the product is positive.
- For \( -2 < x < 5 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 5 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-\infty, -2] \cup [5, +\infty) \)

12. Inequality: \( (x - 1)(x + 1) \leq 0 \)
- Roots: \( x = -1 \) and \( x = 1 \)
- Sign analysis:
- For \( x < -1 \): Both factors are negative, so the product is positive.
- For \( -1 < x < 1 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 1 \): Both factors are positive, so the product is positive.
- Solution: \( x \in [-1, 1] \)

---

#### Row 5:

13. Inequality: \( (2x + 6)(2 - x) \leq 0 \)
- Roots: \( x = -3 \) and \( x = 2 \)
- Sign analysis:
- For \( x < -3 \): Both factors are negative, so the product is positive.
- For \( -3 < x < 2 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 2 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-\infty, -3] \cup [2, +\infty) \)

14. Inequality: \( (x + 1)(8 - 4x) > 0 \)
- Roots: \( x = -1 \) and \( x = 2 \)
- Sign analysis:
- For \( x < -1 \): Both factors are negative, so the product is positive.
- For \( -1 < x < 2 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 2 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-1, 2) \)

15. Inequality: \( (2x - 4)(5 - x) < 0 \)
- Roots: \( x = 2 \) and \( x = 5 \)
- Sign analysis:
- For \( x < 2 \): Both factors are negative, so the product is positive.
- For \( 2 < x < 5 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 5 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (2, 5) \)

---

#### Row 6:

16. Inequality: Find the smallest integer solution of \( (x + 4)(x - 2) \leq 0 \)
- Roots: \( x = -4 \) and \( x = 2 \)
- Sign analysis:
- For \( x < -4 \): Both factors are negative, so the product is positive.
- For \( -4 \leq x \leq 2 \): One factor is positive, and the other is negative, so the product is non-positive.
- For \( x > 2 \): Both factors are positive, so the product is positive.
- Solution: \( x \in [-4, 2] \)
- Smallest integer: \( x = -4 \)

17. Inequality: Find the largest integer solution of \( (x - 3)(x - 5) < 0 \)
- Roots: \( x = 3 \) and \( x = 5 \)
- Sign analysis:
- For \( x < 3 \): Both factors are negative, so the product is positive.
- For \( 3 < x < 5 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 5 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (3, 5) \)
- Largest integer: \( x = 4 \)

18. Inequality: Find the number of integer solutions of \( (x + 4)(x - 2) \leq 0 \)
- From the previous solution, \( x \in [-4, 2] \)
- Integer solutions: \( x = -4, -3, -2, -1, 0, 1, 2 \)
- Number of integer solutions: 7

---

#### Row 7:

19. Inequality: Find the smallest integer solution of \( (x + 4)(x - 2) < 0 \)
- Roots: \( x = -4 \) and \( x = 2 \)
- Sign analysis:
- For \( x < -4 \): Both factors are negative, so the product is positive.
- For \( -4 < x < 2 \): One factor is positive, and the other is negative, so the product is negative.
- For \( x > 2 \): Both factors are positive, so the product is positive.
- Solution: \( x \in (-4, 2) \)
- Smallest integer: \( x = -3 \)

20. Inequality: Find the largest integer solution of \( (x - 3)(x - 5) \leq 0 \)
- Roots: \( x = 3 \) and \( x = 5 \)
- Sign analysis:
- For \( x < 3 \): Both factors are negative, so the product is positive.
- For \( 3 \leq x \leq 5 \): One factor is positive, and the other is negative, so the product is non-positive.
- For \( x > 5 \): Both factors are positive, so the product is positive.
- Solution: \( x \in [3, 5] \)
- Largest integer: \( x = 5 \)

21. Inequality: Find the number of integer solutions of \( (x + 4)(x - 2) < 0 \)
- From the previous solution, \( x \in (-4, 2) \)
- Integer solutions: \( x = -3, -2, -1, 0, 1 \)
- Number of integer solutions: 5

---

Final Answers:

\[
\boxed{
\begin{array}{|c|c|c|}
\hline
x^2 - 5x + 6 \geq 0 & x^2 - x + 2 > 0 & 3x \geq x^2 + 4 \\
x \in (-\infty, 2] \cup [3, +\infty) & x \in (-\infty, +\infty) & x \in \emptyset \\
\hline
2x - 1 \leq (x - 3)(x + 3) & x^2 + 1 \leq 0 & 3x^2 + 12 > 0 \\
x \in (-\infty, -2] \cup [4, +\infty) & x \in \emptyset & x \in (-\infty, +\infty) \\
\hline
\frac{x}{2} + \frac{x(x-1)}{6} \leq 4 & x^2 - 6x + 9 > 0 & x^2 - 4x + 4 \leq 0 \\
x \in [-6, 4] & x \in (-\infty, 3) \cup (3, +\infty) & x = 2 \\
\hline
(x - 1)(x - 3) < 0 & (x + 2)(x - 5) \geq 0 & (x - 1)(x + 1) \leq 0 \\
x \in (1, 3) & x \in (-\infty, -2] \cup [5, +\infty) & x \in [-1, 1] \\
\hline
(2x + 6)(2 - x) \leq 0 & (x + 1)(8 - 4x) > 0 & (2x - 4)(5 - x) < 0 \\
x \in (-\infty, -3] \cup [2, +\infty) & x \in (-1, 2) & x \in (2, 5) \\
\hline
\text{Smallest integer of } (x + 4)(x - 2) \leq 0 & \text{Largest integer of } (x - 3)(x - 5) < 0 & \text{Number of integer solutions of } (x + 4)(x - 2) \leq 0 \\
x = -4 & x = 4 & 7 \\
\hline
\text{Smallest integer of } (x + 4)(x - 2) < 0 & \text{Largest integer of } (x - 3)(x - 5) \leq 0 & \text{Number of integer solutions of } (x + 4)(x - 2) < 0 \\
x = -3 & x = 5 & 5 \\
\hline
\end{array}
}
\]