Solve Inequalities Worksheet: Practice And Solutions
Table of Contents :
When tackling the concept of inequalities, it can sometimes feel overwhelming. However, practice makes perfect! A Solve Inequalities Worksheet is an excellent tool for honing your skills and understanding how to solve various types of inequalities effectively. In this article, we will cover the basics of inequalities, provide practice problems, and explore solutions to help reinforce your understanding.
Understanding Inequalities
Inequalities are mathematical statements that express the relationship between two expressions. Instead of stating that two values are equal (as in an equation), inequalities show that one value is greater than, less than, or not equal to another value. The symbols used in inequalities include:
- Greater than: >
- Less than: <
- Greater than or equal to: ≥
- Less than or equal to: ≤
Types of Inequalities
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Linear Inequalities: Involves variables raised only to the first power. For example:
- (2x + 3 > 7)
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Quadratic Inequalities: Involves variables raised to the second power. For example:
- (x^2 - 4x < 0)
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Rational Inequalities: Involves fractions that have variables in the denominator. For example:
- (\frac{1}{x} > 3)
Practice Problems
Now that we understand the basics of inequalities, let’s dive into some practice problems. Remember to solve each inequality step by step.
Linear Inequalities
- (3x - 5 < 10)
- (-2x + 4 ≥ 0)
- (5 - 3x > 2)
Quadratic Inequalities
- (x^2 - 5x + 6 ≤ 0)
- (x^2 + 2x > 8)
Rational Inequalities
- (\frac{x - 1}{x + 2} ≤ 0)
- (\frac{2x + 3}{x - 1} > 0)
Solutions to Practice Problems
Now let’s go through the solutions to the problems presented above.
Solutions for Linear Inequalities
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Problem: (3x - 5 < 10)
Solution:
- Add 5 to both sides: [ 3x < 15 ]
- Divide by 3: [ x < 5 ]
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Problem: (-2x + 4 ≥ 0)
Solution:
- Subtract 4 from both sides: [ -2x ≥ -4 ]
- Divide by -2 (remember to flip the inequality): [ x ≤ 2 ]
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Problem: (5 - 3x > 2)
Solution:
- Subtract 5 from both sides: [ -3x > -3 ]
- Divide by -3 (flip the inequality): [ x < 1 ]
Solutions for Quadratic Inequalities
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Problem: (x^2 - 5x + 6 ≤ 0)
Solution:
- Factor: [ (x - 2)(x - 3) ≤ 0 ]
- The solutions are (x = 2) and (x = 3).
- Test intervals: ( (-∞, 2], [2, 3], [3, ∞) )
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Problem: (x^2 + 2x > 8)
Solution:
- Rearrange: [ x^2 + 2x - 8 > 0 ]
- Factor: [ (x - 2)(x + 4) > 0 ]
- Analyze the intervals: ( (-∞, -4), (-4, 2), (2, ∞) )
Solutions for Rational Inequalities
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Problem: (\frac{x - 1}{x + 2} ≤ 0)
Solution:
- Determine when the numerator is zero: (x - 1 = 0 \Rightarrow x = 1)
- Determine when the denominator is zero: (x + 2 = 0 \Rightarrow x = -2)
- Analyze sign changes around (x = -2) and (x = 1).
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Problem: (\frac{2x + 3}{x - 1} > 0)
Solution:
- Identify critical points: (2x + 3 = 0 \Rightarrow x = -\frac{3}{2}) and (x - 1 = 0 \Rightarrow x = 1)
- Analyze intervals and determine signs of each.
Summary Table of Solutions
<table> <tr> <th>Problem</th> <th>Solution</th> </tr> <tr> <td>1: (3x - 5 < 10)</td> <td>(x < 5)</td> </tr> <tr> <td>2: (-2x + 4 ≥ 0)</td> <td>(x ≤ 2)</td> </tr> <tr> <td>3: (5 - 3x > 2)</td> <td>(x < 1)</td> </tr> <tr> <td>4: (x^2 - 5x + 6 ≤ 0)</td> <td>(2 ≤ x ≤ 3)</td> </tr> <tr> <td>5: (x^2 + 2x > 8)</td> <td>(x < -4 \text{ or } x > 2)</td> </tr> <tr> <td>6: (\frac{x - 1}{x + 2} ≤ 0)</td> <td>(x \leq 1 \text{ and } x \neq -2)</td> </tr> <tr> <td>7: (\frac{2x + 3}{x - 1} > 0)</td> <td>(x < -\frac{3}{2} \text{ or } x > 1)</td> </tr> </table>
Important Notes
Remember: When dividing or multiplying by a negative number, always flip the inequality sign! This is a common mistake that can lead to incorrect solutions. 🛑
By practicing problems like these, you can enhance your skills in solving inequalities. Keep this worksheet handy for review, and soon you’ll be solving inequalities with ease! Happy studying! 📚✨