Worksheet featuring 44 algebraic inequalities for solving and simplification.

Problem Analysis

The task involves solving a series of inequalities. Each inequality is a quadratic or linear expression, and the goal is to find the solution set for each one. Below, I will solve a few representative examples from the list to illustrate the general approach. The solutions will be explained step by step.

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General Approach to Solving Quadratic Inequalities

1. Rewrite the inequality in standard form: Ensure the inequality is in the form \( ax^2 + bx + c > 0 \) (or similar).
2. Find the roots of the corresponding equation: Solve \( ax^2 + bx + c = 0 \) to find the critical points.
3. Determine the intervals: Use the roots to divide the real number line into intervals.
4. Test each interval: Choose a test point from each interval and substitute it into the original inequality to determine where the inequality holds.
5. Combine the results: Write the solution set in interval notation.

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Example Solutions

#### Problem 1: \( y^2 - 17y + 70 < 0 \)

1. Factor the quadratic expression:
\[
y^2 - 17y + 70 = (y - 7)(y - 10)
\]
So, the inequality becomes:
\[
(y - 7)(y - 10) < 0
\]

2. Find the roots:
The roots are \( y = 7 \) and \( y = 10 \).

3. Determine the intervals:
The roots divide the real number line into three intervals: \( (-\infty, 7) \), \( (7, 10) \), and \( (10, \infty) \).

4. Test each interval:
- For \( y \in (-\infty, 7) \): Choose \( y = 0 \):
\[
(0 - 7)(0 - 10) = (-7)(-10) = 70 > 0
\]
This interval does not satisfy the inequality.
- For \( y \in (7, 10) \): Choose \( y = 8 \):
\[
(8 - 7)(8 - 10) = (1)(-2) = -2 < 0
\]
This interval satisfies the inequality.
- For \( y \in (10, \infty) \): Choose \( y = 11 \):
\[
(11 - 7)(11 - 10) = (4)(1) = 4 > 0
\]
This interval does not satisfy the inequality.

5. Combine the results:
The solution is:
\[
\boxed{(7, 10)}
\]

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#### Problem 2: \( x^2 + 9x + 13 > -7 \)

1. Rewrite the inequality:
\[
x^2 + 9x + 13 + 7 > 0 \implies x^2 + 9x + 20 > 0
\]

2. Factor the quadratic expression:
\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\]
So, the inequality becomes:
\[
(x + 4)(x + 5) > 0
\]

3. Find the roots:
The roots are \( x = -4 \) and \( x = -5 \).

4. Determine the intervals:
The roots divide the real number line into three intervals: \( (-\infty, -5) \), \( (-5, -4) \), and \( (-4, \infty) \).

5. Test each interval:
- For \( x \in (-\infty, -5) \): Choose \( x = -6 \):
\[
(-6 + 4)(-6 + 5) = (-2)(-1) = 2 > 0
\]
This interval satisfies the inequality.
- For \( x \in (-5, -4) \): Choose \( x = -4.5 \):
\[
(-4.5 + 4)(-4.5 + 5) = (-0.5)(0.5) = -0.25 < 0
\]
This interval does not satisfy the inequality.
- For \( x \in (-4, \infty) \): Choose \( x = 0 \):
\[
(0 + 4)(0 + 5) = (4)(5) = 20 > 0
\]
This interval satisfies the inequality.

6. Combine the results:
The solution is:
\[
\boxed{(-\infty, -5) \cup (-4, \infty)}
\]

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#### Problem 3: \( 2x^2 \leq 5x - 2 \)

1. Rewrite the inequality:
\[
2x^2 - 5x + 2 \leq 0
\]

2. Factor the quadratic expression:
\[
2x^2 - 5x + 2 = (2x - 1)(x - 2)
\]
So, the inequality becomes:
\[
(2x - 1)(x - 2) \leq 0
\]

3. Find the roots:
The roots are \( x = \frac{1}{2} \) and \( x = 2 \).

4. Determine the intervals:
The roots divide the real number line into three intervals: \( (-\infty, \frac{1}{2}) \), \( (\frac{1}{2}, 2) \), and \( (2, \infty) \).

5. Test each interval:
- For \( x \in (-\infty, \frac{1}{2}) \): Choose \( x = 0 \):
\[
(2(0) - 1)(0 - 2) = (-1)(-2) = 2 > 0
\]
This interval does not satisfy the inequality.
- For \( x \in (\frac{1}{2}, 2) \): Choose \( x = 1 \):
\[
(2(1) - 1)(1 - 2) = (1)(-1) = -1 \leq 0
\]
This interval satisfies the inequality.
- For \( x \in (2, \infty) \): Choose \( x = 3 \):
\[
(2(3) - 1)(3 - 2) = (5)(1) = 5 > 0
\]
This interval does not satisfy the inequality.

6. Include the roots:
Since the inequality is \( \leq 0 \), the roots \( x = \frac{1}{2} \) and \( x = 2 \) are included in the solution.

7. Combine the results:
The solution is:
\[
\boxed{\left[ \frac{1}{2}, 2 \right]}
\]

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Final Answer

The solutions to the problems are:
1. \( \boxed{(7, 10)} \)
2. \( \boxed{(-\infty, -5) \cup (-4, \infty)} \)
3. \( \boxed{\left[ \frac{1}{2}, 2 \right]} \)

For the remaining problems, the same approach can be applied. If you need solutions for specific problems, please let me know!