Solve The Following Inequality:$-4x + 14 \leq 54$

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear inequality, which is a statement that compares two expressions using a mathematical relationship. We will use the given inequality $-4x + 14 \leq 54$ as an example to demonstrate the step-by-step process of solving linear inequalities.

What are Linear Inequalities?

A linear inequality is a mathematical statement that compares two expressions using a mathematical relationship, such as greater than, less than, greater than or equal to, or less than or equal to. Linear inequalities can be written in the form of $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Types of Linear Inequalities

There are two main types of linear inequalities: strict inequalities and non-strict inequalities.

• Strict Inequalities: These are inequalities that use the symbols $>$ or $<$, such as $2x + 3 > 5$.
• Non-Strict Inequalities: These are inequalities that use the symbols $\geq$ or $\leq$, such as $2x + 3 \leq 5$.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable $x$ on one side of the inequality sign. We can do this by performing algebraic operations, such as addition, subtraction, multiplication, and division, to both sides of the inequality.

Step 1: Subtract 14 from Both Sides

The first step in solving the inequality $-4x + 14 \leq 54$ is to subtract 14 from both sides of the inequality. This will help us isolate the term with the variable $x$.

$$-4x + 14 - 14 \leq 54 - 14$$

$$-4x \leq 40$$

Step 2: Divide Both Sides by -4

The next step is to divide both sides of the inequality by -4. However, when we divide or multiply both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

$$\frac{-4x}{-4} \geq \frac{40}{-4}$$

$$x \geq -10$$

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The solution to the inequality $-4x + 14 \leq 54$ is $x \geq -10$. In interval notation, this can be written as $[-10, \infty)$.

Conclusion

Solving linear inequalities requires a step-by-step approach, and it is essential to follow the correct order of operations to isolate the variable $x$ on one side of the inequality sign. By subtracting 14 from both sides, dividing both sides by -4, and writing the solution in interval notation, we can solve the inequality $-4x + 14 \leq 54$.

Real-World Applications

Linear inequalities have numerous real-world applications in fields such as economics, finance, and engineering. For example, a company may want to determine the maximum amount of money it can spend on a project, given a certain budget constraint. In this case, the company can use linear inequalities to model the relationship between the cost of the project and the available budget.

Tips and Tricks

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

• Use the correct order of operations: When solving linear inequalities, it is essential to follow the correct order of operations, which is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction (PEMDAS).
• Reverse the direction of the inequality sign: When dividing or multiplying both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.
• Check your solution: After solving the inequality, it is essential to check your solution by plugging in a value of $x$ that satisfies the inequality.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear inequalities:

• Not following the correct order of operations: Failing to follow the correct order of operations can lead to incorrect solutions.
• Not reversing the direction of the inequality sign: Failing to reverse the direction of the inequality sign when dividing or multiplying both sides by a negative number can lead to incorrect solutions.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Conclusion

Introduction

In our previous article, we discussed the basics of solving linear inequalities, including the step-by-step process and real-world applications. In this article, we will provide a Q&A guide to help you better understand and master the concept of solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is a mathematical statement that compares two expressions using a mathematical relationship, such as greater than, less than, greater than or equal to, or less than or equal to. Linear inequalities can be written in the form of $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the two main types of linear inequalities?

A: The two main types of linear inequalities are:

• Strict Inequalities: These are inequalities that use the symbols $>$ or $<$, such as $2x + 3 > 5$.
• Non-Strict Inequalities: These are inequalities that use the symbols $\geq$ or $\leq$, such as $2x + 3 \leq 5$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ on one side of the inequality sign. You can do this by performing algebraic operations, such as addition, subtraction, multiplication, and division, to both sides of the inequality.

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

A: The correct order of operations when solving linear inequalities is:

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: What happens when I divide or multiply both sides of an inequality by a negative number?

A: When you divide or multiply both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: How do I write the solution to a linear inequality in interval notation?

A: To write the solution to a linear inequality in interval notation, you need to use the following format:

• Closed Interval: If the inequality is of the form $ax + b \leq c$, the solution is written as $[a, b]$.
• Open Interval: If the inequality is of the form $ax + b < c$, the solution is written as $(a, b)$.
• Half-Open Interval: If the inequality is of the form $ax + b \geq c$, the solution is written as $[a, b)$ or $(a, b]$.

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

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

• Not following the correct order of operations: Failing to follow the correct order of operations can lead to incorrect solutions.
• Not reversing the direction of the inequality sign: Failing to reverse the direction of the inequality sign when dividing or multiplying both sides by a negative number can lead to incorrect solutions.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Q: How can I practice solving linear inequalities?

A: You can practice solving linear inequalities by:

• Working through examples: Try solving linear inequalities using different variables and coefficients.
• Using online resources: There are many online resources available that provide practice problems and exercises for solving linear inequalities.
• Seeking help from a tutor or teacher: If you are struggling to understand or solve linear inequalities, consider seeking help from a tutor or teacher.

Conclusion

Solving linear inequalities is a crucial skill for students to master, and it has numerous real-world applications in fields such as economics, finance, and engineering. By following the correct order of operations, reversing the direction of the inequality sign, and checking the solution, we can solve linear inequalities with confidence.