Solving Inequalities with Addition and Subtraction

Oct 13, 2025 by miodeya.com 135 views

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. One-step inequalities are a type of inequality that involves a single operation, such as addition, subtraction, multiplication, or division. In this article, we will focus on solving one-step inequalities, which will help you develop a strong foundation in algebra and problem-solving skills.

What are One-Step Inequalities?

One-step inequalities are inequalities that involve a single operation. They can be written in the form of:

$ax \leq b$
$ax \geq b$
$ax < b$
$ax > b$

where $a$ and $b$ are constants, and $x$ is the variable.

Solving One-Step Inequalities

Solving Inequalities with Addition and Subtraction

When solving inequalities with addition and subtraction, we need to isolate the variable $x$ on one side of the inequality.

Example 1: Solving $x - 8 \leq -15$

To solve this inequality, we need to add 8 to both sides of the inequality.

$x - 8 + 8 \leq -15 + 8$

This simplifies to:

$x \leq -7$

Therefore, the solution to this inequality is $x \leq -7$ .

Example 2: Solving $-13x < -26$

To solve this inequality, we need to divide both sides of the inequality by -13.

$\frac{-13x}{-13} > \frac{-26}{-13}$

This simplifies to:

$x > 2$

Therefore, the solution to this inequality is $x > 2$ .

Solving Inequalities with Multiplication and Division

When solving inequalities with multiplication and division, we need to isolate the variable $x$ on one side of the inequality.

Example 3: Solving $x - 5 \leq 9$

To solve this inequality, we need to add 5 to both sides of the inequality.

$x - 5 + 5 \leq 9 + 5$

This simplifies to:

$x \leq 14$

Therefore, the solution to this inequality is $x \leq 14$ .

Example 4: Solving $-9 + x < -28$

To solve this inequality, we need to add 9 to both sides of the inequality.

$-9 + x + 9 < -28 + 9$

This simplifies to:

$x < -19$

Therefore, the solution to this inequality is $x < -19$ .

Solving Inequalities with Fractions

When solving inequalities with fractions, we need to isolate the variable $x$ on one side of the inequality.

Example 5: Solving $-2x \leq -20$

To solve this inequality, we need to divide both sides of the inequality by -2.

$\frac{-2x}{-2} \leq \frac{-20}{-2}$

This simplifies to:

$x \leq 10$

Therefore, the solution to this inequality is $x \leq 10$ .

Example 6: Solving $\frac{x}{6} \geq -2$

To solve this inequality, we need to multiply both sides of the inequality by 6.

$\frac{x}{6} \times 6 \geq -2 \times 6$

This simplifies to:

$x \geq -12$

Therefore, the solution to this inequality is $x \geq -12$ .

Conclusion

Solving one-step inequalities is an essential skill in mathematics that helps you develop problem-solving skills and algebraic thinking. By following the steps outlined in this article, you can solve various types of one-step inequalities, including those with addition and subtraction, multiplication and division, and fractions. Remember to isolate the variable $x$ on one side of the inequality and simplify the expression to find the solution.

Practice Problems

Solve the inequality $x + 3 \leq 7$ .
Solve the inequality $-4x > -12$ .
Solve the inequality $x - 2 \geq 5$ .
Solve the inequality $-3 + x < -2$ .
Solve the inequality $\frac{x}{3} \leq 4$ .

Answer Key

$x \leq 4$
$x < 3$
$x \geq 7$
$x < 1$
$x \leq 12$

Introduction

In our previous article, we discussed how to solve one-step inequalities, which are a fundamental concept in mathematics. However, we understand that sometimes, you may have questions or need clarification on certain topics. In this article, we will address some of the most frequently asked questions about one-step inequalities.

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

A: A one-step inequality involves a single operation, such as addition, subtraction, multiplication, or division, whereas a multi-step inequality involves multiple operations.

Q: How do I solve an inequality with a negative coefficient?

A: When solving an inequality with a negative coefficient, you need to multiply both sides of the inequality by the negative coefficient. This will change the direction of the inequality.

Example: Solve the inequality $-3x \leq 12$

To solve this inequality, you need to multiply both sides of the inequality by -3.

$\frac{-3x}{-3} \geq \frac{12}{-3}$

This simplifies to:

$x \geq -4$

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.

Example: Solve the inequality $x + 2 \leq 5$

To solve this inequality, you can subtract 2 from both sides of the inequality.

$x + 2 - 2 \leq 5 - 2$

This simplifies to:

$x \leq 3$

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

A: When solving an inequality with a fraction, you need to multiply both sides of the inequality by the reciprocal of the fraction.

Example: Solve the inequality $\frac{x}{2} \geq 3$

To solve this inequality, you need to multiply both sides of the inequality by 2.

$\frac{x}{2} \times 2 \geq 3 \times 2$

This simplifies to:

$x \geq 6$

Q: Can I multiply or divide both sides of an inequality by a negative number?

A: Yes, you can multiply or divide both sides of an inequality by a negative number. However, you need to change the direction of the inequality.

Example: Solve the inequality $-2x \leq 10$

To solve this inequality, you need to divide both sides of the inequality by -2.

$\frac{-2x}{-2} \geq \frac{10}{-2}$

This simplifies to:

$x \geq -5$

Q: How do I know which direction to change the inequality when multiplying or dividing by a negative number?

A: When multiplying or dividing both sides of an inequality by a negative number, you need to change the direction of the inequality. To determine which direction to change, you can use the following rule:

If you multiply both sides of the inequality by a negative number, change the direction of the inequality.
If you divide both sides of the inequality by a negative number, change the direction of the inequality.

Conclusion

We hope this Q&A article has helped you clarify any questions you may have had about one-step inequalities. Remember to practice solving inequalities with different operations and coefficients to develop your problem-solving skills. If you have any further questions or need additional clarification, feel free to ask.

Practice Problems

Solve the inequality $-4x \geq 12$
Solve the inequality $\frac{x}{3} \leq 2$
Solve the inequality $x - 2 \geq 5$
Solve the inequality $-3 + x < -2$
Solve the inequality $\frac{x}{2} \geq 4$

Answer Key

$x \leq -3$
$x \leq 6$
$x \geq 7$
$x < 1$
$x \geq 8$