Solve The Inequality:$\[ 3(x + 7) \ \textless \ \frac{x}{2} + 1 \\]

Feb 27, 2025 by ADMIN 70 views

Solving Inequalities: A Step-by-Step Guide to Solving the Inequality $3(x + 7) < \frac{x}{2} + 1$

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, but one is either greater than or less than the other. In this article, we will focus on solving the inequality $3(x + 7) < \frac{x}{2} + 1$ . We will break down the solution into step-by-step instructions, making it easy to understand and follow.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it means. The inequality $3(x + 7) < \frac{x}{2} + 1$ states that the expression $3(x + 7)$ is less than the expression $\frac{x}{2} + 1$ . To solve this inequality, we need to isolate the variable $x$ .

Step 1: Distribute the 3 to the terms inside the parentheses

The first step in solving the inequality is to distribute the 3 to the terms inside the parentheses. This will give us $3x + 21 < \frac{x}{2} + 1$ .

Step 2: Subtract 21 from both sides of the inequality

Next, we need to subtract 21 from both sides of the inequality to get rid of the constant term on the left-hand side. This gives us $3x < \frac{x}{2} + 1 - 21$ .

Step 3: Simplify the right-hand side of the inequality

Now, let's simplify the right-hand side of the inequality by combining the constants. This gives us $3x < \frac{x}{2} - 20$ .

Step 4: Subtract $\frac{x}{2}$ from both sides of the inequality

To isolate the variable $x$ , we need to subtract $\frac{x}{2}$ from both sides of the inequality. This gives us $3x - \frac{x}{2} < -20$ .

Step 5: Simplify the left-hand side of the inequality

Now, let's simplify the left-hand side of the inequality by combining the like terms. This gives us $\frac{6x - x}{2} < -20$ .

Step 6: Simplify the fraction

Next, let's simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 1. This gives us $\frac{5x}{2} < -20$ .

Step 7: Multiply both sides of the inequality by 2

To get rid of the fraction, we need to multiply both sides of the inequality by 2. This gives us $5x < -40$ .

Step 8: Divide both sides of the inequality by 5

Finally, we need to divide both sides of the inequality by 5 to isolate the variable $x$ . This gives us $x < -8$ .

Conclusion

In conclusion, the solution to the inequality $3(x + 7) < \frac{x}{2} + 1$ is $x < -8$ . This means that any value of $x$ that is less than -8 will satisfy the inequality.

Graphing the Solution

To visualize the solution, we can graph the inequality on a number line. The number line will have a closed circle at -8, indicating that -8 is not included in the solution. The number line will also have an open circle at -8, indicating that any value of $x$ that is less than -8 is included in the solution.

Real-World Applications

Inequalities have many real-world applications, including:

Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid:

Not following the order of operations: When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that the correct solution is obtained.
Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.
Not considering the direction of the inequality: Failing to consider the direction of the inequality can lead to incorrect solutions.

Conclusion

In conclusion, solving inequalities requires a step-by-step approach, careful attention to detail, and a thorough understanding of the concepts involved. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy.
Solving Inequalities: A Q&A Guide

In our previous article, we discussed how to solve the inequality $3(x + 7) < \frac{x}{2} + 1$ . In this article, we will answer some frequently asked questions about solving inequalities.

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

A: An inequality is a statement that two expressions are not equal, but one is either greater than or less than the other. An equation, on the other hand, is a statement that two expressions are equal.

Q: How do I know which direction to use when solving an inequality?

A: When solving an inequality, you need to determine the direction of the inequality. If the inequality is of the form $ax < b$ , then the direction is "less than". If the inequality is of the form $ax > b$ , then the direction is "greater than".

Q: What is the order of operations when solving inequalities?

A: When solving inequalities, you need to follow the order of operations (PEMDAS):

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I isolate the variable when solving an inequality?

A: To isolate the variable when solving an inequality, you need to perform the following steps:

Add or subtract the same value to both sides: Add or subtract the same value to both sides of the inequality to get rid of any constants.
Multiply or divide both sides by the same value: Multiply or divide both sides of the inequality by the same value to get rid of any fractions.
Simplify the inequality: Simplify the inequality by combining like terms.

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

A: A linear inequality is an inequality that can be written in the form $ax < b$ , where $a$ and $b$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c < 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to follow these steps:

Factor the quadratic expression: Factor the quadratic expression into the product of two binomials.
Set each factor equal to zero: Set each factor equal to zero and solve for the variable.
Graph the solution: Graph the solution on a number line.
Determine the direction of the inequality: Determine the direction of the inequality based on the graph.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{ax}{b} < c$ , where $a$ , $b$ , and $c$ are constants. A polynomial inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c < 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to follow these steps:

Factor the numerator and denominator: Factor the numerator and denominator into the product of two binomials.
Set each factor equal to zero: Set each factor equal to zero and solve for the variable.
Graph the solution: Graph the solution on a number line.
Determine the direction of the inequality: Determine the direction of the inequality based on the graph.

Conclusion