Which Equation Is True For X 6 And X 2

Which Equation Is True for x = 6 and x = 2? A Deep Dive into Equation Solving

This article explores the fascinating world of equations and how to determine which equation holds true for specific values of x, specifically x = 6 and x = 2. We will delve into various types of equations – linear, quadratic, and even those involving absolute values – and demonstrate how to test whether a given equation is satisfied when these values are substituted for x. We'll also cover strategies to solve for x and understand the implications of different solution sets.

Understanding the Basics: Variables and Equations

Before we jump into complex equations, let's establish a firm foundation. An equation is a mathematical statement asserting that two expressions are equal. It contains an equals sign (=), separating the left-hand side (LHS) from the right-hand side (RHS). A variable, often represented by x (or other letters), is a symbol that represents an unknown value. Our goal is to find the value(s) of x that make the equation true.

Example: A Simple Linear Equation

Consider the equation: x + 3 = 5

This is a linear equation because the highest power of x is 1. To solve this, we need to isolate x. Subtracting 3 from both sides gives:

x = 5 - 3

x = 2

Therefore, this equation is true only when x = 2. If we substitute any other value for x, the equation will be false.

Testing Equations for x = 6 and x = 2

Let's move on to more complex scenarios. We'll test different equations to see if they hold true when x = 6 and x = 2.

1. Linear Equations

Equation 1: 2x + 4 = 16

• For x = 6: 2(6) + 4 = 12 + 4 = 16. True!
• For x = 2: 2(2) + 4 = 4 + 4 = 8. False!

Equation 2: x - 4 = 2

• For x = 6: 6 - 4 = 2. True!
• For x = 2: 2 - 4 = -2. False!

Equation 3: 5x - 10 = 20

• For x = 6: 5(6) - 10 = 30 - 10 = 20. True!
• For x = 2: 5(2) - 10 = 10 - 10 = 0. False!

Conclusion: Notice that none of these linear equations are true for both x = 6 and x = 2. This highlights the fact that different equations have different solution sets.

2. Quadratic Equations

Quadratic equations are of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations can have up to two solutions.

Equation 4: x² - 8x + 12 = 0

This equation can be factored as (x - 6)(x - 2) = 0. This means the solutions are x = 6 and x = 2.

• For x = 6: (6)² - 8(6) + 12 = 36 - 48 + 12 = 0. True!
• For x = 2: (2)² - 8(2) + 12 = 4 - 16 + 12 = 0. True!

Conclusion: This quadratic equation is true for both x = 6 and x = 2. This is because these values are the roots (solutions) of the equation.

3. Equations with Absolute Values

Absolute value equations involve the absolute value function, denoted by |x|, which gives the distance of x from zero. For example, |3| = 3 and |-3| = 3.

Equation 5: |x - 4| = 2

This equation means the distance between x and 4 is 2.

• For x = 6: |6 - 4| = |2| = 2. True!
• For x = 2: |2 - 4| = |-2| = 2. True!

Conclusion: This absolute value equation is true for both x = 6 and x = 2.

4. Equations with Higher Powers

Let's explore an equation with a higher power of x.

Equation 6: x³ - 10x² + 24x = 0

This equation can be factored as x(x - 6)(x - 4) = 0. The solutions are x = 0, x = 6, and x = 4.

• For x = 6: (6)³ - 10(6)² + 24(6) = 216 - 360 + 144 = 0. True!
• For x = 2: (2)³ - 10(2)² + 24(2) = 8 - 40 + 48 = 16. False!

Solving Equations: Techniques and Strategies

Solving equations involves manipulating the equation to isolate the variable. Here's a brief overview of common techniques:

• Linear Equations: Use inverse operations (addition/subtraction, multiplication/division) to isolate x.
• Quadratic Equations: Factoring, the quadratic formula, or completing the square can be used to find the solutions.
• Absolute Value Equations: Consider two cases: the expression inside the absolute value is positive, and the expression is negative.
• Higher-Order Equations: Factoring, using the rational root theorem, or numerical methods might be necessary.

Implications of Different Solution Sets

The number of solutions an equation has depends on its type and complexity.

• Linear equations: Typically have one solution.
• Quadratic equations: Can have two, one, or no real solutions.
• Higher-order equations: Can have multiple solutions.

Understanding the solution set is crucial for interpreting the equation's meaning within its context (e.g., in physics, engineering, or economics).

Conclusion: Choosing the Right Equation

Determining which equation is true for specific values of x involves careful substitution and, if necessary, solving the equation. The examples above demonstrate that different equation types have different solution sets. The choice of the "correct" equation depends entirely on the specific problem and the desired solution set. Linear equations offer simple, single-solution scenarios, while quadratic and higher-order equations provide opportunities for more complex analysis and multiple solutions. Understanding these nuances is essential for mastering algebraic problem-solving. This article has provided a foundation for approaching various types of equations and recognizing the implications of their respective solution sets. Remember to always check your work and use appropriate techniques to solve for the unknown variable (x).