1) Solve 3x 8-6. Graph The Solution.A)B)C)D)

1) Solve 3x 8-6. Graph The Solution.A)B)C)D)

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Introduction

Mathematics is a fundamental skill that helps us understand the world around us. One common type of problem involves solving equations and graphing solutions to visualize relationships. In this article, we will explore how to solve the equation 3x + 8 - 6 and how to accurately graph the solution. Additionally, we will analyze multiple-choice options labeled A), B), C), and D), providing detailed explanations to determine the correct answer.

Whether you're a student brushing up on algebra or someone interested in enhancing your problem-solving skills, this comprehensive guide will walk you through each step with clarity. Let's begin by understanding the problem statement.

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Understanding the Problem: What Are We Solving?

The problem statement is:

Solve 3x + 8 - 6. Graph the Solution. A) B) C) D)

At first glance, it appears incomplete because it doesn't explicitly specify an equality (e.g., equals zero or another number). However, in many algebra problems, an expression like 3x + 8 - 6 is meant to be set equal to a value, often zero, to solve for x.

Assumption for Clarity

For the purpose of this tutorial, we'll assume the problem is:

Solve for x in the equation:

\[ 3x + 8 - 6 = 0 \]

which simplifies to:

\[ 3x + 2 = 0 \]

If the original problem intended a different equation, the method would be similar, and the steps can be adjusted accordingly.

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Step 1: Simplify the Expression

Before solving, simplify the algebraic expression:

\[
3x + 8 - 6
\]

Calculate the constants:

\[
8 - 6 = 2
\]

So, the simplified expression becomes:

\[
3x + 2
\]

Now, the equation to solve is:

\[
3x + 2 = 0
\]

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Step 2: Solve the Equation for x

To find the value of x, follow these algebraic steps:

1. Subtract 2 from both sides:

\[
3x = -2
\]

1. Divide both sides by 3:

Solution:

\[
x = -\frac{2}{3}
\]

This is the exact solution to the equation.

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Step 3: Graphting the Solution

Graphing helps visualize solutions to equations, especially in coordinate systems. Since the solution involves a single value of x, we can plot this on a number line or a Cartesian coordinate plane.

Graphing on a Number Line

• Draw a horizontal line.


• Mark the point x = -2/3 on the number line.


• Use a solid dot to indicate the solution point.

• Consider the equation as a vertical line where x = -2/3.


• Plot the line x = -2/3, which is a vertical line crossing the x-axis at -2/3.


• This line extends infinitely in the y-direction.

Step 4: Analyzing Multiple Choice Options (A, B, C, D)

The problem references options labeled A), B), C), and D). Typically, these options provide different possible solutions or graphs. Let’s analyze what they might represent:

Possible options:

• A) \( x = -\frac{2}{3} \) — Correct solution


• B) \( x = \frac{2}{3} \) — Incorrect, opposite sign


• C) \( x = 0 \) — Incorrect


• D) No solution — Incorrect, since the solution exists

• Check each option by substituting the value into the original simplified equation \(3x + 2 = 0\).

\[
x = -\frac{2}{3}
\]

Plug into \(3x + 2\):

\[
3 \times -\frac{2}{3} + 2 = -2 + 2 = 0
\]

Result: Valid solution.

Testing option B:

\[
x = \frac{2}{3}
\]

\[
3 \times \frac{2}{3} + 2 = 2 + 2 = 4 \neq 0
\]

Invalid.

Similarly, options C) and D) can be tested, but based on calculations, only A) is correct.

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Step 5: Graphing the Correct Solution

Visual Representation:

• The graph of x = -2/3 is a vertical line crossing the x-axis at -2/3.


• The line extends infinitely in the positive and negative y-directions.


• The point (-2/3, y) for any y-value lies on this line.

1. Draw the coordinate axes.


2. Locate -2/3 on the x-axis.


3. Draw a straight vertical line passing through that point.


4. Label the line as x = -2/3.

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Additional Tips for Solving and Graphing Linear Equations

• Always simplify the algebraic expression before solving.


• Isolate the variable on one side of the equation.


• When graphing, understand whether the solution is a point, line, or region.


• Use graph paper or graphing tools for precise visualization.


• Verify solutions by substitution.

Common Mistakes to Avoid

• Forgetting to simplify the expression first.


• Mixing up the signs when solving for x.


• Not checking the solution against the original equation.


• Mislabeling the graph or plotting the wrong point.

Summary

• The original expression 3x + 8 - 6 simplifies to 3x + 2.


• Solving 3x + 2 = 0 yields x = -2/3.


• The correct option among A), B), C), D) is A).


• Graphing x = -2/3 involves drawing a vertical line at x = -2/3 on the coordinate plane.


• Visualizing solutions helps in understanding the nature of equations and their solutions.

Final Remarks

Mastering the skill of solving algebraic equations and graphing solutions is essential for progressing in mathematics. Practice regularly by tackling different equations and visualizing their solutions. Remember to simplify expressions, check your work, and use graphical representations to deepen your understanding.

If you encounter similar problems, follow the structured approach outlined above to efficiently arrive at the correct solution and visualize it accurately.

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Frequently Asked Questions (FAQs)

Q1: What does solving an equation like 3x + 8 - 6 = 0 tell us?
A: It helps identify the specific value of x that makes the equation true, providing insight into the relationship between variables.

Q2: How does graphing help in understanding solutions?
A: Graphing visualizes solutions, showing where and how the solution exists within the coordinate system, making abstract algebraic solutions more concrete.

Q3: Can the solution be a range?
A: Yes, in inequalities or systems of equations, the solution might be a range or a region, but for a single linear equation like this, it's a specific point or line.

Q4: What should I do if I get a different answer?
A: Double-check your algebraic steps, ensure correct substitution, and verify your graph for accuracy.

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By following these detailed steps and explanations, you can confidently solve similar algebra problems and understand their graphical representations. Happy solving!