Solving For X Mastering Equations And Inequalities

Jul 16, 2025 by ADMIN 51 views

Solving Algebraic Equations and Inequalities: A Comprehensive Guide

This article provides a comprehensive guide on solving for x in various algebraic equations and inequalities. We will cover linear equations, linear inequalities, and quadratic equations, providing step-by-step solutions and explanations to help you master these fundamental concepts in mathematics. Understanding these concepts is crucial for success in algebra and beyond. This article aims to provide a clear and concise approach, ensuring that readers can confidently tackle similar problems.

4.1.1 Solving the Linear Equation: 15 - 3x = 2x - 25

In this section, we will delve into solving the linear equation 15 - 3x = 2x - 25. Linear equations are foundational in algebra, and mastering their solution is crucial for more advanced topics. The key to solving linear equations is to isolate the variable x on one side of the equation. This involves performing algebraic operations such as addition, subtraction, multiplication, and division while maintaining the equation's balance. Our step-by-step approach will ensure clarity and understanding.

Step-by-Step Solution

Combine like terms: To begin, we want to gather all terms containing x on one side of the equation and constant terms on the other side. We can achieve this by adding 3x to both sides of the equation:
```
15 - 3x + 3x = 2x - 25 + 3x
15 = 5x - 25
```
This step eliminates the -3x term from the left side and combines the x terms on the right side, simplifying the equation.
Isolate the x term: Next, we need to isolate the term with x (which is 5x) on the right side. We can do this by adding 25 to both sides of the equation:
```
15 + 25 = 5x - 25 + 25
40 = 5x
```
This step moves the constant term from the right side to the left side, further isolating the x term.
Solve for x: Finally, to solve for x, we need to divide both sides of the equation by the coefficient of x, which is 5:
```
40 / 5 = 5x / 5
8 = x
x = 8
```
This step isolates x, giving us the solution to the equation.

Verification

To ensure our solution is correct, we can substitute x = 8 back into the original equation:

15 - 3(8) = 2(8) - 25
15 - 24 = 16 - 25
-9 = -9

Since both sides of the equation are equal, our solution x = 8 is verified.

Key Takeaways

The key to solving linear equations is to isolate the variable x by performing the same operations on both sides of the equation.
Combining like terms simplifies the equation, making it easier to solve.
Verification is a crucial step to ensure the accuracy of the solution.

By understanding these steps and practicing similar problems, you can confidently solve linear equations.

4.1.2 Solving the Compound Inequality: -3 ≤ 2x - 7 ≤ 7

Moving on to inequalities, this section focuses on solving the compound inequality -3 ≤ 2x - 7 ≤ 7. Compound inequalities involve two inequalities joined by "and" or "or". In this case, we have a three-part inequality, which means the expression 2x - 7 is bounded between -3 and 7. Solving compound inequalities requires applying algebraic operations to all parts of the inequality simultaneously to isolate the variable x. This section will provide a detailed, step-by-step solution to this type of problem.

Step-by-Step Solution

Isolate the x term: Our goal is to isolate x in the middle part of the inequality. To begin, we add 7 to all three parts of the inequality:
```
-3 + 7 ≤ 2x - 7 + 7 ≤ 7 + 7
4 ≤ 2x ≤ 14
```
This step eliminates the -7 from the middle part, bringing us closer to isolating x.
Solve for x: Next, we divide all three parts of the inequality by the coefficient of x, which is 2:
```
4 / 2 ≤ 2x / 2 ≤ 14 / 2
2 ≤ x ≤ 7
```
This step isolates x, giving us the solution to the compound inequality.

Solution Interpretation

The solution 2 ≤ x ≤ 7 means that x is greater than or equal to 2 and less than or equal to 7. In interval notation, this is represented as [2, 7]. This notation indicates that x can take any value within this interval, including the endpoints 2 and 7.

Graphical Representation

We can represent the solution graphically on a number line. We draw a closed circle at 2 and a closed circle at 7, indicating that these values are included in the solution. Then, we shade the region between 2 and 7 to represent all the values of x that satisfy the inequality.

Key Takeaways

Solving compound inequalities involves performing the same operations on all parts of the inequality.
The solution represents a range of values for x.
Interval notation and graphical representation are useful ways to express the solution.

By understanding these steps and practicing with similar inequalities, you can confidently solve compound inequalities.

4.1.3 Solving the Quadratic Equation: (x + 5)(2x - 8) = 0

Finally, we address the quadratic equation (x + 5)(2x - 8) = 0. Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants. One of the most common methods to solve quadratic equations is by factoring, which is the approach we will use here. The zero-product property is crucial in this method, stating that if the product of two factors is zero, then at least one of the factors must be zero. This section will walk through the process of solving this quadratic equation step by step.

Step-by-Step Solution

Apply the zero-product property: The given equation is already factored, which simplifies the process. According to the zero-product property, if (x + 5)(2x - 8) = 0, then either (x + 5) = 0 or (2x - 8) = 0.
Solve for x in each factor: We now have two linear equations to solve:
- x + 5 = 0 Subtract 5 from both sides:
```
x + 5 - 5 = 0 - 5
x = -5
```
- 2x - 8 = 0 Add 8 to both sides:
```
2x - 8 + 8 = 0 + 8
2x = 8
```
  Divide both sides by 2:
```
2x / 2 = 8 / 2
x = 4
```

Solutions

Therefore, the solutions to the quadratic equation are x = -5 and x = 4. These are the values of x that make the equation true.

Verification

To verify our solutions, we substitute each value back into the original equation:

For x = -5:

(-5 + 5)(2(-5) - 8) = 0
(0)(-10 - 8) = 0
0 = 0

The equation holds true.

For x = 4:

(4 + 5)(2(4) - 8) = 0
(9)(8 - 8) = 0
(9)(0) = 0
0 = 0

The equation holds true.

Both solutions are verified.

Key Takeaways

The zero-product property is essential for solving factored quadratic equations.
A quadratic equation can have two distinct solutions, one solution, or no real solutions.
Verification ensures the accuracy of the solutions.

By mastering the factoring method and the zero-product property, you can confidently solve quadratic equations.

In this article, we have explored solving for x in various types of equations and inequalities, including linear equations, compound inequalities, and quadratic equations. We have provided step-by-step solutions and explanations to enhance your understanding of these fundamental algebraic concepts. Remember to practice these techniques regularly to build your skills and confidence in solving mathematical problems. Mastering these concepts is not only crucial for algebra but also for more advanced mathematical studies.