Problem 21 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$x-6=-2$$

Step-by-Step Solution

Answer

The solution is \(x = 4\); verify by substitution and graph it on a number line with a point at 4.

Step 1: Write down the equation

Start with the equation given in the problem: \[ x - 6 = -2 \]

Step 2: Isolate the variable

To solve for \(x\), we need to isolate it on one side of the equation. We can do this by adding 6 to both sides of the equation:\[ x - 6 + 6 = -2 + 6 \]

Step 3: Simplify the equation

The \(-6 + 6\) on the left side cancels out, leaving:\[ x = 4 \]

Step 4: Verify the solution

Substitute \(x = 4\) back into the original equation to ensure it satisfies the equation:\[ 4 - 6 = -2 \]This simplifies to \(-2 = -2\), which is true. Thus, \(x = 4\) is the correct solution.

Step 5: Graph the solution on a number line

Draw a number line and place a closed circle on the number 4, since \(x = 4\) is the exact solution. Representing it on a number line shows the value of \(x\) as a point.

Key Concepts

Checking Solutions

When solving linear equations, it's essential to check your solution to ensure it's correct. After finding a solution, substitute it back into the original equation. For example, in the equation \( x - 6 = -2 \), we found the solution \( x = 4 \). To check if this is correct, substitute 4 back into the equation. This gives us:

• \( 4 - 6 = -2 \)
• Simplify: \( -2 = -2 \)

Since both sides of the equation are equal, the solution \( x = 4 \) is verified as correct. Checking solutions is a critical step to avoid mistakes and ensures the solution satisfies the original equation.

Isolating Variable

Isolating the variable in an equation is a fundamental step for solving it. In the equation \( x - 6 = -2 \), our goal is to solve for \( x \). To do this, we must bring \( x \) by itself on one side of the equality. Adding 6 to both sides accomplishes this:

• Original: \( x - 6 = -2 \)
• Add 6: \( x - 6 + 6 = -2 + 6 \)
• Simplify: \( x = 4 \)

This process "undoes" the subtraction of 6, effectively isolating \( x \). Always perform the same operation to both sides to maintain balance. This principle is key in solving any equation, ensuring the equality stays true.

Number Line Graph

Graphing solutions on a number line provides a visual representation, making it easier to understand where the solution lies. For \( x = 4 \), draw a straight line and mark the value 4 on it.

• Place a closed circle directly on number 4.
• This indicates \( x = 4 \) is a specific, complete solution.

Using a number line is a great way to visualize solutions, particularly helpful for understanding the location and magnitude of numbers. This can be especially beneficial for learners who prefer visual over numerical data.

Equation Verification

Equation verification is about making sure that the calculated solution not only satisfies the problem but also adheres to the logical structure of the equation. After isolating the variable and finding \( x = 4 \), we verify it through both substitution and graphical representation. Dedicate a moment to verify:

• Substitute back into the original equation: \( 4 - 6 = -2 \)
• Ensure that both sides equal: \( -2 = -2 \)

Successful verification confirms the solution is correct and aligned with the equation's logic. This sequential scrutiny adds confidence and accuracy to the result and is a step students shouldn’t skip.