Solving -7 < 3x + 2 ≤ 5 A Step-by-Step Guide

by Admin 45 views

In the realm of mathematics, inequalities play a crucial role in defining ranges and conditions for variables. Among the various types of inequalities, compound inequalities present a unique challenge, requiring a combination of techniques to arrive at the solution. This article delves into the intricacies of solving a specific compound inequality, -7 < 3x + 2 ≤ 5, providing a step-by-step guide and exploring the underlying concepts.

Understanding Compound Inequalities

Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." The inequality -7 < 3x + 2 ≤ 5 is a compound inequality joined by "and," indicating that the solution must satisfy both inequalities simultaneously. To solve such inequalities, we aim to isolate the variable, 'x' in this case, while maintaining the integrity of the inequality signs.

Step-by-Step Solution

To solve the compound inequality -7 < 3x + 2 ≤ 5, we'll follow a systematic approach:

1. Isolate the Variable Term

The primary goal is to isolate the term containing the variable, which is '3x' in our case. We can achieve this by performing the same operation on all parts of the inequality. In this instance, we'll subtract 2 from all three parts:

-7 - 2 < 3x + 2 - 2 ≤ 5 - 2

This simplifies to:

-9 < 3x ≤ 3

2. Isolate the Variable

Now, we need to isolate 'x' completely. Since 'x' is multiplied by 3, we'll divide all parts of the inequality by 3:

-9 / 3 < 3x / 3 ≤ 3 / 3

This gives us:

-3 < x ≤ 1

3. Express the Solution

The solution to the compound inequality is -3 < x ≤ 1. This can be interpreted as "x is greater than -3 and less than or equal to 1." In interval notation, this solution is represented as (-3, 1]. The parenthesis indicates that -3 is not included in the solution set, while the square bracket indicates that 1 is included.

4. Graphical Representation

Visualizing the solution on a number line can provide a clearer understanding. We represent the solution -3 < x ≤ 1 by drawing a number line and marking -3 and 1. We use an open circle at -3 to indicate that it's not included in the solution and a closed circle at 1 to indicate that it is included. The region between -3 and 1 is shaded, representing all the values of 'x' that satisfy the inequality.

Analyzing the Solution Options

Now, let's analyze the given options in light of our solution:

  • A. x > -3 and x ≤ 1: This option accurately reflects our solution. 'x > -3' means x is greater than -3, and 'x ≤ 1' means x is less than or equal to 1. This corresponds to the interval notation (-3, 1].
  • B. x > -3 or x ≤ 1: This option uses the word "or," which means the solution would include any value that satisfies either inequality. This would encompass all real numbers since any number is either greater than -3 or less than or equal to 1. This is not the correct solution for the given compound inequality.
  • C. x > 3 and x ≤ 3: This option is incorrect because it contradicts itself. There is no number that is strictly greater than 3 and simultaneously less than or equal to 3.
  • D. x < -3 or x ≤ 1: This option is also incorrect. 'x < -3' represents values less than -3, and 'x ≤ 1' represents values less than or equal to 1. The "or" conjunction would include all values less than or equal to 1, which does not match our solution.

Therefore, the correct answer is A. x > -3 and x ≤ 1.

Key Concepts and Principles

Several key concepts and principles underpin the process of solving compound inequalities:

1. Maintaining Inequality

When performing operations on an inequality, it's crucial to maintain the integrity of the inequality sign. Adding or subtracting the same value from all parts of the inequality does not change the inequality sign. However, multiplying or dividing by a negative number reverses the inequality sign.

2. "And" vs. "Or"

The words "and" and "or" have distinct meanings in the context of compound inequalities. An "and" compound inequality requires that all inequalities be satisfied simultaneously. An "or" compound inequality requires that at least one of the inequalities be satisfied.

3. Interval Notation

Interval notation is a concise way to represent the solution set of an inequality. Parentheses indicate that the endpoint is not included, while square brackets indicate that the endpoint is included. For example, (a, b) represents all values between a and b, excluding a and b. [a, b] represents all values between a and b, including a and b. (a, ∞) represents all values greater than a, and (-∞, b) represents all values less than b.

4. Number Line Representation

Representing the solution on a number line provides a visual aid for understanding the solution set. Open circles are used to indicate endpoints that are not included, and closed circles are used to indicate endpoints that are included. The shaded region represents the values that satisfy the inequality.

Common Mistakes to Avoid

When solving compound inequalities, it's essential to be aware of common mistakes:

1. Forgetting to Apply Operations to All Parts

When performing operations on a compound inequality, ensure that you apply the operation to all parts of the inequality, not just some of them. This maintains the balance of the inequality.

2. Incorrectly Handling Negative Numbers

Remember that multiplying or dividing by a negative number reverses the inequality sign. Failing to do so will lead to an incorrect solution.

3. Misinterpreting "And" and "Or"

Understand the difference between "and" and "or" conjunctions. An "and" compound inequality requires all inequalities to be satisfied, while an "or" compound inequality requires at least one inequality to be satisfied.

4. Incorrectly Representing the Solution

Pay attention to whether the endpoints should be included or excluded in the solution set. Use parentheses for endpoints that are not included and square brackets for endpoints that are included.

Applications of Compound Inequalities

Compound inequalities have numerous applications in various fields, including:

1. Physics

In physics, compound inequalities can be used to define the range of values for physical quantities such as temperature, velocity, and pressure. For instance, the temperature of a substance might need to be within a specific range for a chemical reaction to occur.

2. Engineering

In engineering, compound inequalities can be used to set constraints on the design and operation of systems. For example, the voltage in an electrical circuit might need to be within a certain range to prevent damage to components.

3. Economics

In economics, compound inequalities can be used to model economic relationships and predict outcomes. For instance, the price of a product might need to be within a certain range to maximize profits.

4. Computer Science

In computer science, compound inequalities can be used to define conditions for program execution and data validation. For example, the age of a user might need to be within a certain range to access certain features of a website.

Conclusion

Solving compound inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the underlying concepts, following a systematic approach, and avoiding common mistakes, you can confidently tackle these types of problems. The compound inequality -7 < 3x + 2 ≤ 5 serves as a valuable example, illustrating the steps involved in isolating the variable and expressing the solution. Remember to pay close attention to the meaning of "and" and "or," the rules for manipulating inequalities, and the proper representation of the solution set.