Start by simplifying the inequality

Oct 13, 2025 by talynt.org 48 views

Solving Inequalities: A Step-by-Step Guide to Understanding and Solving the Inequality $\frac{5x + 6}{2} \leq 4x + 2$

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 $\frac{5x + 6}{2} \leq 4x + 2$ . We will break down the solution into manageable steps, making it easy to understand and follow.

Before we dive into solving the inequality, let's first understand what it means. The given inequality is $\frac{5x + 6}{2} \leq 4x + 2$ . This means that the expression on the left-hand side, $\frac{5x + 6}{2}$ , is less than or equal to the expression on the right-hand side, $4x + 2$ . Our goal is to find the values of $x$ that satisfy this inequality.

Step 1: Multiply Both Sides by 2

To simplify the inequality, we can start by multiplying both sides by 2. This will eliminate the fraction and make it easier to work with.

\frac{5x + 6}{2} \leq 4x + 2

\Rightarrow 5x + 6 \leq 2(4x + 2)

\Rightarrow 5x + 6 \leq 8x + 4

Step 2: Subtract 5x from Both Sides

Next, we can subtract $5x$ from both sides to isolate the variable $x$ .

5x + 6 \leq 8x + 4

\Rightarrow 6 \leq 3x + 4

Step 3: Subtract 4 from Both Sides

Now, we can subtract 4 from both sides to further isolate the variable $x$ .

6 \leq 3x + 4

\Rightarrow 2 \leq 3x

Step 4: Divide Both Sides by 3

Finally, we can divide both sides by 3 to solve for $x$ .

2 \leq 3x

\Rightarrow \frac{2}{3} \leq x

In conclusion, we have successfully solved the inequality $\frac{5x + 6}{2} \leq 4x + 2$ . By following the steps outlined above, we found that the solution to the inequality is $x \geq \frac{2}{3}$ . This means that any value of $x$ greater than or equal to $\frac{2}{3}$ will satisfy the inequality.

Solving inequalities has numerous real-world applications. For example, in finance, inequalities can be used to determine the minimum or maximum value of an investment. In engineering, inequalities can be used to design and optimize systems. In medicine, inequalities can be used to model and analyze the spread of diseases.

Here are some tips and tricks to help you solve inequalities:

Start by simplifying the inequality: Before solving the inequality, try to simplify it by combining like terms or eliminating fractions.
Use inverse operations: To isolate the variable, use inverse operations such as addition, subtraction, multiplication, or division.
Check your solution: Once you have solved the inequality, check your solution by plugging in values of $x$ to see if they satisfy the inequality.

Here are some common mistakes to avoid when solving inequalities:

Not simplifying the inequality: Failing to simplify the inequality can make it difficult to solve.
Not using inverse operations: Failing to use inverse operations can make it difficult to isolate the variable.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined above and avoiding common mistakes, you can successfully solve inequalities and apply them to real-world problems. Remember to start by simplifying the inequality, use inverse operations, and check your solution. With practice and patience, you will become proficient in solving inequalities and be able to apply them to a wide range of problems.
Solving Inequalities: A Q&A Guide

In our previous article, we discussed how to solve the inequality $\frac{5x + 6}{2} \leq 4x + 2$ . We broke down the solution into manageable steps, making it easy to understand and follow. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is an inequality?

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

Q: How do I know which direction to solve the inequality?

A: To determine which direction to solve the inequality, you need to look at the inequality sign. If the sign is $\leq$ or $\geq$ , you will solve for the values of $x$ that make the expression on the left-hand side less than or equal to the expression on the right-hand side. If the sign is $>$ or $<$ , you will solve for the values of $x$ that make the expression on the left-hand side greater than or less than the expression on the right-hand side.

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 $ax + b \leq cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Factor the quadratic expression: If possible, factor the quadratic expression to make it easier to work with.
Use the sign chart method: Create a sign chart to determine the intervals where the quadratic expression is positive or negative.
Determine the solution: Based on the sign chart, determine the intervals where the quadratic expression is less than or equal to zero.

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 $\frac{ax + b}{cx + d} \leq 0$ , where $a$ , $b$ , $c$ , and $d$ are constants. A polynomial inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

Factor the numerator and denominator: If possible, factor the numerator and denominator to make it easier to work with.
Use the sign chart method: Create a sign chart to determine the intervals where the rational expression is positive or negative.
Determine the solution: Based on the sign chart, determine the intervals where the rational expression is less than or equal to zero.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Not simplifying the inequality: Failing to simplify the inequality can make it difficult to solve.
Not using inverse operations: Failing to use inverse operations can make it difficult to isolate the variable.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

If you are looking for additional resources to help you learn how to solve inequalities, here are some suggestions:

Textbooks: There are many textbooks available that cover the topic of inequalities, including "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
Online resources: There are many online resources available that provide tutorials and examples on how to solve inequalities, including Khan Academy and Mathway.
Practice problems: Practice problems are an essential part of learning how to solve inequalities. You can find practice problems in textbooks, online resources, or by creating your own problems.