1 5 Practice Solving Inequalities

Mastering Inequalities: 15 Practice Problems to Sharpen Your Skills

Solving inequalities is a crucial skill in algebra and beyond, forming the bedrock for understanding many mathematical concepts. This comprehensive guide provides 15 practice problems of varying difficulty, designed to build your confidence and mastery of inequality solving techniques. We'll cover a range of inequality types, from simple linear inequalities to those involving absolute values and quadratic expressions. By the end, you’ll be well-equipped to tackle any inequality challenge thrown your way. Whether you're preparing for an exam, brushing up on your skills, or simply seeking a deeper understanding, this guide is your perfect companion.

Understanding the Basics of Inequalities

Before we dive into the practice problems, let's refresh our understanding of inequalities. Inequalities are mathematical statements comparing two expressions that are not necessarily equal. They use the following symbols:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

The key principle in solving inequalities is to isolate the variable, similar to solving equations. However, there’s a crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is a common source of errors, so pay close attention to this rule.

15 Practice Problems: A Graded Approach

The problems are categorized for clarity, progressing from simpler to more complex scenarios. Each problem includes a detailed solution to guide your learning.

Section 1: Linear Inequalities

These problems involve solving inequalities with a single variable raised to the power of one.

Problem 1: Solve 3x + 5 > 11

Solution:

1. Subtract 5 from both sides: 3x > 6
2. Divide both sides by 3: x > 2

Problem 2: Solve 7 - 2y ≤ 1

Solution:

1. Subtract 7 from both sides: -2y ≤ -6
2. Divide both sides by -2 (and remember to reverse the inequality sign!): y ≥ 3

Problem 3: Solve 4(x - 2) ≥ 2x + 6

Solution:

1. Distribute the 4: 4x - 8 ≥ 2x + 6
2. Subtract 2x from both sides: 2x - 8 ≥ 6
3. Add 8 to both sides: 2x ≥ 14
4. Divide both sides by 2: x ≥ 7

Section 2: Compound Inequalities

These problems involve two or more inequalities combined using "and" or "or."

Problem 4: Solve -3 < 2x + 1 < 7

Solution:

This is a compound inequality. We solve it by isolating x in the middle:

1. Subtract 1 from all parts: -4 < 2x < 6
2. Divide all parts by 2: -2 < x < 3

Problem 5: Solve x + 4 > 10 or x - 2 < -5

Solution:

Solve each inequality separately:

• x + 4 > 10 => x > 6
• x - 2 < -5 => x < -3

The solution is x > 6 or x < -3.

Section 3: Inequalities with Absolute Values

Absolute value inequalities require careful consideration of both positive and negative cases.

Problem 6: Solve |x - 3| < 2

Solution:

This inequality means that the distance between x and 3 is less than 2. We can rewrite this as a compound inequality:

-2 < x - 3 < 2

1. Add 3 to all parts: 1 < x < 5

Problem 7: Solve |2x + 1| ≥ 5

Solution:

This inequality means that the distance between 2x + 1 and 0 is greater than or equal to 5. This leads to two separate inequalities:

• 2x + 1 ≥ 5 => 2x ≥ 4 => x ≥ 2
• 2x + 1 ≤ -5 => 2x ≤ -6 => x ≤ -3

The solution is x ≥ 2 or x ≤ -3.

Section 4: Quadratic Inequalities

These problems involve inequalities with a variable raised to the power of two. They often require factoring or the quadratic formula.

Problem 8: Solve x² - 4x + 3 > 0

Solution:

1. Factor the quadratic: (x - 1)(x - 3) > 0
2. Find the roots: x = 1 and x = 3
3. Test intervals: The parabola opens upwards, so the inequality is true when x < 1 or x > 3.

Problem 9: Solve x² + 2x - 8 ≤ 0

Solution:

1. Factor the quadratic: (x + 4)(x - 2) ≤ 0
2. Find the roots: x = -4 and x = 2
3. Test intervals: The parabola opens upwards, so the inequality is true when -4 ≤ x ≤ 2.

Section 5: More Complex Inequalities

These problems combine techniques from previous sections.

Problem 10: Solve |x - 1| + 2 < 5

Solution:

1. Subtract 2 from both sides: |x - 1| < 3
2. Rewrite as a compound inequality: -3 < x - 1 < 3
3. Add 1 to all parts: -2 < x < 4

Problem 11: Solve (x + 2)(x - 3)² > 0

Solution:

Notice that (x-3)² is always non-negative. Therefore, the inequality is satisfied when x+2 >0 and (x-3)²≠0. Hence, the solution is x > -2 and x ≠ 3.

Problem 12: Solve 2|x + 4| - 3 > 5

Solution:

1. Add 3 to both sides: 2|x + 4| > 8
2. Divide by 2: |x + 4| > 4
3. This gives two inequalities: x + 4 > 4 or x + 4 < -4
4. Solve each: x > 0 or x < -8

Section 6: Inequalities with Fractions

These problems involve inequalities containing fractions.

Problem 13: Solve (x + 1)/(x - 2) < 0

Solution:

1. Find the critical points: x = -1 and x = 2
2. Test intervals: The expression changes sign at these points. Testing values in each interval shows that the inequality is satisfied when -1 < x < 2.

Problem 14: Solve (2x - 1) / (x + 3) ≥ 1

Solution:

1. Subtract 1 from both sides: (2x - 1)/(x + 3) - 1 ≥ 0
2. Find a common denominator: (2x - 1 - (x + 3))/(x + 3) ≥ 0
3. Simplify: (x - 4)/(x + 3) ≥ 0
4. Find critical points: x = 4 and x = -3
5. Test intervals: The inequality is satisfied when x ≤ -3 or x ≥ 4.

Section 7: Applications of Inequalities

Inequalities appear in various real-world problems.

Problem 15: A rectangular garden must have an area of at least 50 square meters. If the length is 10 meters, what must the width be?

Solution:

Let w be the width. The area is given by A = 10w. We have the inequality 10w ≥ 50. Dividing by 10, we get w ≥ 5 meters.

Conclusion

Solving inequalities is a multi-faceted skill demanding careful attention to detail and a clear understanding of the fundamental rules. By working through these 15 practice problems, you've gained valuable experience in various inequality types and techniques. Remember the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number. Regular practice is key to mastering inequalities, so continue practicing with varied problems to build your confidence and proficiency. Remember to always double-check your work and understand the meaning of your solution within the context of the problem. With consistent effort, you'll become adept at tackling even the most challenging inequality problems.