determine if a number is a solution or an inequality

Sembilsg

2 min read

·

21-01-2025

74

0

Share

Inequalities, unlike equations, don't have a single solution. Instead, they represent a range of values that satisfy the given condition. This article will guide you through the process of determining whether a specific number falls within this solution set. We'll cover various types of inequalities and offer clear examples.

Understanding Inequalities

Before we dive into determining solutions, let's quickly recap the basics of inequalities. Inequalities use symbols like:

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

These symbols express a relationship between two expressions, indicating one is smaller, larger, or equal to the other. For example, x > 5 means "x is greater than 5."

How to Determine if a Number is a Solution

The core process involves substituting the number into the inequality and evaluating the resulting statement. If the statement is true, the number is a solution; if false, it isn't.

Step-by-Step Guide

1. Substitute: Replace the variable in the inequality with the number you're testing.
2. Simplify: Perform any necessary calculations to simplify the expression.
3. Evaluate: Determine if the resulting statement is true or false.

Example 1: Is x = 7 a solution to the inequality x > 5?

1. Substitute: 7 > 5
2. Simplify: The expression is already simplified.
3. Evaluate: 7 is greater than 5. This statement is true. Therefore, x = 7 is a solution.

Example 2: Is x = 2 a solution to the inequality x ≥ 5?

1. Substitute: 2 ≥ 5
2. Simplify: The expression is already simplified.
3. Evaluate: 2 is not greater than or equal to 5. This statement is false. Therefore, x = 2 is not a solution.

Example 3: Is x = -3 a solution to the inequality 2x + 5 < 1?

1. Substitute: 2(-3) + 5 < 1
2. Simplify: -6 + 5 < 1 which simplifies to -1 < 1
3. Evaluate: -1 is less than 1. This statement is true. Therefore, x = -3 is a solution.

Dealing with Compound Inequalities

Compound inequalities involve multiple inequalities joined by "and" or "or."

Example 4: Is x = 3 a solution to the compound inequality 1 < x < 5?

This means x must be greater than 1 AND less than 5.

1. Substitute: 1 < 3 < 5
2. Evaluate: 3 is greater than 1 AND less than 5. This statement is true. Therefore, x = 3 is a solution.

Example 5: Is x = 7 a solution to the compound inequality x < 5 OR x > 10?

1. Substitute: 7 < 5 OR 7 > 10
2. Evaluate: 7 is not less than 5. 7 is not greater than 10. Both parts of the compound inequality are false. For an "OR" statement to be true, at least one part must be true. Since neither is true, this statement is false. Therefore, x = 7 is not a solution.

Graphical Representation

Visualizing inequalities on a number line can be helpful. Solutions are represented by the shaded regions. A closed circle (•) indicates inclusion (≤ or ≥), while an open circle (◦) indicates exclusion (< or >).

Solving Inequalities (A Brief Overview)

While this article focuses on checking solutions, understanding how to solve inequalities is crucial. The process is similar to solving equations, but with one key difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Conclusion

Determining if a number is a solution to an inequality is a fundamental skill in algebra. By systematically substituting, simplifying, and evaluating, you can confidently determine whether a given value satisfies the conditions of the inequality. Remember to pay close attention to the inequality symbols and the rules for compound inequalities. Mastering this skill lays the groundwork for more advanced algebraic concepts.