Solving Inequalities with Addition or Subtraction

Oct 13, 2025 by miodeya.com 66 views

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that two expressions are not equal, and it can be either greater than, less than, greater than or equal to, or less than or equal to. In this article, we will focus on solving inequalities in terms of intervals and illustrating the solution set on the real number line. We will cover four different types of inequalities and provide step-by-step solutions.

Solving Inequalities: A Step-by-Step Approach

Solving Inequalities with Addition or Subtraction

When solving inequalities, we can add or subtract the same value to both sides of the inequality without changing the solution set. Let's consider the following inequality:

11. $2x + 7 \ \textgreater \ 3$

To solve this inequality, we need to isolate the variable x. We can start by subtracting 7 from both sides of the inequality:

2x + 7 - 7 \  \textgreater \  3 - 7

This simplifies to:

2x \  \textgreater \  -4

Next, we can divide both sides of the inequality by 2 to solve for x:

\frac{2x}{2} \  \textgreater \  \frac{-4}{2}

This simplifies to:

x \  \textgreater \  -2

Therefore, the solution set for this inequality is x > -2.

Solving Inequalities with Multiplication or Division

When solving inequalities, we can multiply or divide both sides of the inequality by the same non-zero value without changing the solution set. Let's consider the following inequality:

12. $4 - 3x \ \textgreater \ 6$

To solve this inequality, we need to isolate the variable x. We can start by subtracting 4 from both sides of the inequality:

4 - 4 - 3x \  \textgreater \  6 - 4

This simplifies to:

-3x \  \textgreater \  2

Next, we can divide both sides of the inequality by -3 to solve for x. However, we need to be careful when dividing by a negative number, as it will change the direction of the inequality:

\frac{-3x}{-3} \  \textless \  \frac{2}{-3}

This simplifies to:

x \  \textless \  -\frac{2}{3}

Therefore, the solution set for this inequality is x < -2/3.

Solving Linear Equations

When solving linear equations, we can add or subtract the same value to both sides of the equation without changing the solution set. Let's consider the following equation:

13. $1 - x = 2$

To solve this equation, we need to isolate the variable x. We can start by subtracting 1 from both sides of the equation:

1 - 1 - x = 2 - 1

This simplifies to:

-x = 1

Next, we can multiply both sides of the equation by -1 to solve for x:

-x \times -1 = 1 \times -1

This simplifies to:

x = -1

Therefore, the solution set for this equation is x = -1.

Solving Linear Inequalities with Absolute Value

When solving linear inequalities with absolute value, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. Let's consider the following inequality:

14. $1 + 5r \ \textgreater \ 5$

To solve this inequality, we need to isolate the variable r. We can start by subtracting 1 from both sides of the inequality:

1 + 1 + 5r \  \textgreater \  5 - 1

This simplifies to:

5r \  \textgreater \  4

Next, we can divide both sides of the inequality by 5 to solve for r:

\frac{5r}{5} \  \textgreater \  \frac{4}{5}

This simplifies to:

r \  \textgreater \  \frac{4}{5}

Therefore, the solution set for this inequality is r > 4/5.

Conclusion

In this article, we have covered four different types of inequalities and provided step-by-step solutions. We have learned how to solve inequalities with addition or subtraction, multiplication or division, linear equations, and linear inequalities with absolute value. We have also learned how to illustrate the solution set on the real number line. By following these steps, we can solve a wide range of inequalities and gain a deeper understanding of mathematical concepts.

Real Number Line

The real number line is a visual representation of the set of all real numbers. It is a line that extends infinitely in both directions, with each point on the line representing a real number. The real number line is a useful tool for illustrating the solution set of an inequality.

Solution Set

The solution set of an inequality is the set of all values that satisfy the inequality. It is the set of all real numbers that make the inequality true. The solution set can be represented on the real number line as a shaded region or an interval.

Interval Notation

Interval notation is a way of representing a solution set using intervals. It is a way of writing the solution set in a concise and clear manner. Interval notation is commonly used in mathematics to represent the solution set of an inequality.

Common Intervals

There are several common intervals that are used to represent the solution set of an inequality. These include:

(-∞, a): This interval represents all real numbers less than a.
(a, ∞): This interval represents all real numbers greater than a.
(-∞, a]: This interval represents all real numbers less than or equal to a.
[a, ∞): This interval represents all real numbers greater than or equal to a.

Conclusion

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal. It can be either greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can add or subtract the same value to both sides of the inequality without changing the solution set. You can also multiply or divide both sides of the inequality by the same non-zero value without changing the solution set.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that two expressions are equal. An inequality is a statement that two expressions are not equal.

Q: How do I know which direction to change the inequality sign when multiplying or dividing both sides 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 sign. For example, if the inequality is x > 2 and you multiply both sides by -1, the inequality becomes x < -2.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values that satisfy the inequality. It is the set of all real numbers that make the inequality true.

Q: How do I represent the solution set of an inequality on the real number line?

A: You can represent the solution set of an inequality on the real number line as a shaded region or an interval.

Q: What is interval notation?

A: Interval notation is a way of representing a solution set using intervals. It is a way of writing the solution set in a concise and clear manner.

Q: What are the common intervals used to represent the solution set of an inequality?

A: The common intervals used to represent the solution set of an inequality are:

(-∞, a): This interval represents all real numbers less than a.
(a, ∞): This interval represents all real numbers greater than a.
(-∞, a]: This interval represents all real numbers less than or equal to a.
[a, ∞): This interval represents all real numbers greater than or equal to a.

Q: How do I solve an inequality with absolute value?

A: To solve an inequality with absolute value, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Q: What is the difference between a linear equation and a linear inequality?

A: A linear equation is a statement that two expressions are equal. A linear inequality is a statement that two expressions are not equal.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can add or subtract the same value to both sides of the equation without changing the solution set.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can add or subtract the same value to both sides of the inequality without changing the solution set. You can also multiply or divide both sides of the inequality by the same non-zero value without changing the solution set.

Conclusion

In conclusion, solving inequalities is an important concept in mathematics that requires a deep understanding of mathematical concepts. By following the steps outlined in this article, you can solve a wide range of inequalities and gain a deeper understanding of mathematical concepts. We have learned how to solve inequalities with addition or subtraction, multiplication or division, linear equations, and linear inequalities with absolute value. We have also learned how to illustrate the solution set on the real number line and use interval notation to represent the solution set.