SOLVING ONE-STEP INEQUALITIES1) $x + 20 \ \textgreater \ 25$2) $x - 8 \leq -15$ 3) $-15x \ \textgreater \ -45$4) $-13x \ \textless \ -26$5) $\frac{x}{2} \ \textgreater \ 18$6) $\frac{x}{-5} \leq

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Introduction

One-step inequalities are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will delve into the world of one-step inequalities, exploring the different types of inequalities, and providing step-by-step solutions to various examples. By the end of this article, you will be equipped with the knowledge and skills to tackle even the most challenging one-step inequalities.

What are One-Step Inequalities?

One-step inequalities are mathematical statements that compare two expressions, with the goal of finding the solution set that satisfies the inequality. In other words, one-step inequalities are equations with a missing value, and the goal is to isolate the variable on one side of the inequality sign.

Types of One-Step Inequalities

There are several types of one-step inequalities, including:

  • Linear inequalities: These are inequalities that involve a linear expression, such as x + 20 > 25.
  • Rational inequalities: These are inequalities that involve a rational expression, such as x/2 > 18.
  • Absolute value inequalities: These are inequalities that involve an absolute value expression, such as |x| > 3.

Solving Linear One-Step Inequalities

Linear one-step inequalities are the most common type of inequality. To solve a linear one-step inequality, follow these steps:

  1. Isolate the variable: Move all terms containing the variable to one side of the inequality sign.
  2. Add or subtract the same value: Add or subtract the same value to both sides of the inequality to isolate the variable.
  3. Multiply or divide by a positive value: Multiply or divide both sides of the inequality by a positive value to solve for the variable.

Example 1: Solving x + 20 > 25

To solve the inequality x + 20 > 25, follow these steps:

  1. Subtract 20 from both sides: x + 20 - 20 > 25 - 20
  2. Simplify: x > 5

Solving Rational One-Step Inequalities

Rational one-step inequalities involve a rational expression. To solve a rational one-step inequality, follow these steps:

  1. Isolate the variable: Move all terms containing the variable to one side of the inequality sign.
  2. Multiply or divide by a positive value: Multiply or divide both sides of the inequality by a positive value to solve for the variable.

Example 2: Solving x/2 > 18

To solve the inequality x/2 > 18, follow these steps:

  1. Multiply both sides by 2: (x/2) × 2 > 18 × 2
  2. Simplify: x > 36

Solving Absolute Value One-Step Inequalities

Absolute value one-step inequalities involve an absolute value expression. To solve an absolute value one-step inequality, follow these steps:

  1. Isolate the variable: Move all terms containing the variable to one side of the inequality sign.
  2. Solve for the positive and negative cases: Solve for the positive case (x > a) and the negative case (x < -a).

Example 3: Solving |x| > 3

To solve the inequality |x| > 3, follow these steps:

  1. Solve for the positive case: x > 3
  2. Solve for the negative case: x < -3

Conclusion

Solving one-step inequalities is a crucial skill for students to master. By understanding the different types of one-step inequalities and following the step-by-step solutions provided in this article, you will be equipped with the knowledge and skills to tackle even the most challenging one-step inequalities. Remember to isolate the variable, add or subtract the same value, and multiply or divide by a positive value to solve for the variable.

Practice Problems

Try solving the following one-step inequalities:

  1. x - 8 ≤ -15
  2. -15x > -45
  3. -13x < -26
  4. x/2 > 18
  5. x/-5 ≤ 3

Answer Key

  1. x ≤ -7
  2. x < 3
  3. x > 2
  4. x > 36
  5. x ≥ -15
    One-Step Inequality Q&A: Frequently Asked Questions =====================================================

Introduction

In our previous article, we explored the world of one-step inequalities, covering the different types of inequalities and providing step-by-step solutions to various examples. However, we know that practice makes perfect, and sometimes, it's helpful to have a quick reference guide to common questions and answers. In this article, we'll address some of the most frequently asked questions about one-step inequalities.

Q: What is the difference between a one-step inequality and a two-step inequality?

A: A one-step inequality is an inequality that can be solved in one step, whereas a two-step inequality requires two steps to solve. For example, the inequality x + 20 > 25 is a one-step inequality, while the inequality x + 20 > 25 - 10 is a two-step inequality.

Q: How do I know which direction to move the inequality sign when solving an inequality?

A: When solving an inequality, you can use the following rule: if you multiply or divide both sides of the inequality by a positive value, the inequality sign remains the same. However, if you multiply or divide both sides of the inequality by a negative value, the inequality sign is reversed.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. This is a fundamental property of inequalities, and it allows you to isolate the variable on one side of the inequality sign.

Q: How do I solve an inequality with a fraction?

A: To solve an inequality with a fraction, follow these steps:

  1. Multiply or divide both sides of the inequality by the denominator of the fraction.
  2. Simplify the inequality.
  3. Solve for the variable.

Q: Can I use the same steps to solve an absolute value inequality as I would a linear inequality?

A: No, you cannot use the same steps to solve an absolute value inequality as you would a linear inequality. Absolute value inequalities require a different approach, and you must consider both the positive and negative cases.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, substitute a value into the inequality and check if it is true or false. For example, if you have the inequality x > 5, you can substitute x = 6 and check if 6 > 5 is true.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, be sure to check your calculator's settings and ensure that it is set to solve inequalities correctly.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, follow these steps:

  1. Determine the solution set of the inequality.
  2. Plot the solution set on a number line.
  3. Use a closed circle to indicate the endpoint of the solution set, and an open circle to indicate the endpoint of the solution set.

Conclusion

We hope this Q&A article has provided you with a better understanding of one-step inequalities and has addressed some of the most frequently asked questions. Remember to practice solving one-step inequalities and to use the step-by-step solutions provided in our previous article as a reference guide.

Practice Problems

Try solving the following one-step inequalities:

  1. x - 8 ≤ -15
  2. -15x > -45
  3. -13x < -26
  4. x/2 > 18
  5. x/-5 ≤ 3

Answer Key

  1. x ≤ -7
  2. x < 3
  3. x > 2
  4. x > 36
  5. x ≥ -15