Inequalities Worksheet featuring ten algebraic problems to solve, including linear and quadratic inequalities.

Let's solve each inequality step by step.

---

Problem 1: \( 6 - b < 8 - 4b \)

1. Start with the given inequality:
\[
6 - b < 8 - 4b
\]

2. Add \( 4b \) to both sides to eliminate the \( -4b \) on the right:
\[
6 - b + 4b < 8 - 4b + 4b
\]
Simplify:
\[
6 + 3b < 8
\]

3. Subtract 6 from both sides to isolate the term with \( b \):
\[
6 + 3b - 6 < 8 - 6
\]
Simplify:
\[
3b < 2
\]

4. Divide both sides by 3:
\[
b < \frac{2}{3}
\]

Solution:
\[
\boxed{b < \frac{2}{3}}
\]

---

Problem 2: \( 4(x + \frac{1}{2}) - 2(x + \frac{3}{2}) \leq 5 \)

1. Distribute the constants inside the parentheses:
\[
4(x + \frac{1}{2}) = 4x + 2
\]
\[
-2(x + \frac{3}{2}) = -2x - 3
\]
So the inequality becomes:
\[
4x + 2 - 2x - 3 \leq 5
\]

2. Combine like terms:
\[
(4x - 2x) + (2 - 3) \leq 5
\]
Simplify:
\[
2x - 1 \leq 5
\]

3. Add 1 to both sides:
\[
2x - 1 + 1 \leq 5 + 1
\]
Simplify:
\[
2x \leq 6
\]

4. Divide both sides by 2:
\[
x \leq 3
\]

Solution:
\[
\boxed{x \leq 3}
\]

---

Problem 3: \( -1 < x + 2 < 5 \)

This is a compound inequality. We can solve it in two parts simultaneously.

1. Subtract 2 from all three parts:
\[
-1 - 2 < x + 2 - 2 < 5 - 2
\]
Simplify:
\[
-3 < x < 3
\]

Solution:
\[
\boxed{-3 < x < 3}
\]

---

Problem 4: \( 3(y + 5) \leq 2(y + 1) \)

1. Distribute the constants inside the parentheses:
\[
3(y + 5) = 3y + 15
\]
\[
2(y + 1) = 2y + 2
\]
So the inequality becomes:
\[
3y + 15 \leq 2y + 2
\]

2. Subtract \( 2y \) from both sides:
\[
3y + 15 - 2y \leq 2y + 2 - 2y
\]
Simplify:
\[
y + 15 \leq 2
\]

3. Subtract 15 from both sides:
\[
y + 15 - 15 \leq 2 - 15
\]
Simplify:
\[
y \leq -13
\]

Solution:
\[
\boxed{y \leq -13}
\]

---

Problem 5: \( -5(u - 19) \leq -6 + 2u \)

1. Distribute the \(-5\) inside the parentheses:
\[
-5(u - 19) = -5u + 95
\]
So the inequality becomes:
\[
-5u + 95 \leq -6 + 2u
\]

2. Add \( 5u \) to both sides:
\[
-5u + 95 + 5u \leq -6 + 2u + 5u
\]
Simplify:
\[
95 \leq -6 + 7u
\]

3. Add 6 to both sides:
\[
95 + 6 \leq -6 + 7u + 6
\]
Simplify:
\[
101 \leq 7u
\]

4. Divide both sides by 7:
\[
\frac{101}{7} \leq u
\]
or
\[
u \geq \frac{101}{7}
\]

Solution:
\[
\boxed{u \geq \frac{101}{7}}
\]

---

Problem 6: \( 3 - 2(n - 4) > -1 \)

1. Distribute the \(-2\) inside the parentheses:
\[
-2(n - 4) = -2n + 8
\]
So the inequality becomes:
\[
3 - 2n + 8 > -1
\]

2. Combine like terms:
\[
3 + 8 - 2n > -1
\]
Simplify:
\[
11 - 2n > -1
\]

3. Subtract 11 from both sides:
\[
11 - 2n - 11 > -1 - 11
\]
Simplify:
\[
-2n > -12
\]

4. Divide both sides by \(-2\) (and reverse the inequality sign):
\[
n < 6
\]

Solution:
\[
\boxed{n < 6}
\]

---

Problem 7: \( 4r + 3 > 2r + 11 \)

1. Subtract \( 2r \) from both sides:
\[
4r + 3 - 2r > 2r + 11 - 2r
\]
Simplify:
\[
2r + 3 > 11
\]

2. Subtract 3 from both sides:
\[
2r + 3 - 3 > 11 - 3
\]
Simplify:
\[
2r > 8
\]

3. Divide both sides by 2:
\[
r > 4
\]

Solution:
\[
\boxed{r > 4}
\]

---

Problem 8: \( (x - 5)(x - 2) \leq 0 \)

This is a quadratic inequality. First, find the roots of the equation \( (x - 5)(x - 2) = 0 \):
\[
x - 5 = 0 \quad \text{or} \quad x - 2 = 0
\]
\[
x = 5 \quad \text{or} \quad x = 2
\]

The roots divide the number line into three intervals: \( (-\infty, 2) \), \( (2, 5) \), and \( (5, \infty) \). Test a point in each interval:

1. For \( x \in (-\infty, 2) \), choose \( x = 0 \):
\[
(0 - 5)(0 - 2) = (-5)(-2) = 10 > 0
\]
(Not part of the solution.)

2. For \( x \in (2, 5) \), choose \( x = 3 \):
\[
(3 - 5)(3 - 2) = (-2)(1) = -2 \leq 0
\]
(Part of the solution.)

3. For \( x \in (5, \infty) \), choose \( x = 6 \):
\[
(6 - 5)(6 - 2) = (1)(4) = 4 > 0
\]
(Not part of the solution.)

The inequality \( (x - 5)(x - 2) \leq 0 \) holds when \( x \) is in the interval \( [2, 5] \).

Solution:
\[
\boxed{[2, 5]}
\]

---

Problem 9: \( x^2 + 7x + 10 < 10 \)

1. Subtract 10 from both sides:
\[
x^2 + 7x + 10 - 10 < 10 - 10
\]
Simplify:
\[
x^2 + 7x < 0
\]

2. Factor the left-hand side:
\[
x(x + 7) < 0
\]

The roots of the equation \( x(x + 7) = 0 \) are:
\[
x = 0 \quad \text{or} \quad x = -7
\]

The roots divide the number line into three intervals: \( (-\infty, -7) \), \( (-7, 0) \), and \( (0, \infty) \). Test a point in each interval:

1. For \( x \in (-\infty, -7) \), choose \( x = -8 \):
\[
(-8)(-8 + 7) = (-8)(-1) = 8 > 0
\]
(Not part of the solution.)

2. For \( x \in (-7, 0) \), choose \( x = -1 \):
\[
(-1)(-1 + 7) = (-1)(6) = -6 < 0
\]
(Part of the solution.)

3. For \( x \in (0, \infty) \), choose \( x = 1 \):
\[
(1)(1 + 7) = (1)(8) = 8 > 0
\]
(Not part of the solution.)

The inequality \( x(x + 7) < 0 \) holds when \( x \) is in the interval \( (-7, 0) \).

Solution:
\[
\boxed{(-7, 0)}
\]

---

Problem 10: \( \frac{x^2 + 6}{2} \geq 53 \)

1. Multiply both sides by 2 to eliminate the fraction:
\[
x^2 + 6 \geq 106
\]

2. Subtract 6 from both sides:
\[
x^2 + 6 - 6 \geq 106 - 6
\]
Simplify:
\[
x^2 \geq 100
\]

3. Take the square root of both sides (remembering to consider both positive and negative roots):
\[
|x| \geq 10
\]

This means:
\[
x \leq -10 \quad \text{or} \quad x \geq 10
\]

Solution:
\[
\boxed{(-\infty, -10] \cup [10, \infty)}
\]

---

Final Answers:

1. \( \boxed{b < \frac{2}{3}} \)
2. \( \boxed{x \leq 3} \)
3. \( \boxed{-3 < x < 3} \)
4. \( \boxed{y \leq -13} \)
5. \( \boxed{u \geq \frac{101}{7}} \)
6. \( \boxed{n < 6} \)
7. \( \boxed{r > 4} \)
8. \( \boxed{[2, 5]} \)
9. \( \boxed{(-7, 0)} \)
10. \( \boxed{(-\infty, -10] \cup [10, \infty)} \)