Problem 10 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$4(2 x-3)=32$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5.5\).

Step 1: Distribute

First, distribute \(4\) inside the parenthesis to the terms \(2x - 3\) which gives us \(8x - 12 =32\).

Step 2: Add 12 to both sides

Add \(12\) to both sides of the equation to isolate the term with \(x\). This leads to \(8x = 44\).

Step 3: Division

Divide both sides of the equation by \(8\) to solve for \(x\). This results in \(x = 44 / 8 = 5.5\).

Step 4: Check the solution

Substitute \(5.5\) for \(x\) in the original equation: \(4(2*5.5 - 3) = 4(8) = 32\) which is equal to the right side, hence the solution verifies.

Key Concepts

Equation Distribution

When solving equations that involve parentheses, such as \(4(2x - 3) = 32\), the first step is often to eliminate the parentheses. This is where equation distribution comes into play. The goal of distribution is to simplify the equation, making it easier to solve.
Distributing involves multiplying the term outside of the parentheses by each term inside. In this case, we distribute the \(4\) to both \(2x\) and \(-3\). This gives us:

• \(4 \times 2x = 8x\)
• \(4 \times -3 = -12\)

Putting it all together, the distributed form of \(4(2x - 3)\) is \(8x - 12\). Now the equation is much simpler: \(8x - 12 = 32\). Next, we will learn about isolating variables to continue solving from here.

Isolating Variables

After distributing and simplifying the equation to \(8x - 12 = 32\), the next step in solving linear equations is to isolate the variable. Isolating the variable means getting \(x\) by itself on one side of the equation.

To start, we need to move the \(-12\) to the other side to further simplify the equation. This is done by adding \(12\) to both sides. Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced.

• \(8x - 12 + 12 = 32 + 12\)
• This simplifies to \(8x = 44\)

Now that we have \(8x = 44\), we can isolate \(x\) by dividing both sides by \(8\). This results in:

• \(x = \frac{44}{8} = 5.5\)

With the variable \(x\) isolated and solved, the equation is much simpler to understand. Next, we will check our solution.

Checking Solutions

Once we have determined that \(x = 5.5\), it is very important to check the solution. Checking ensures that there were no mistakes made during the solving process and verifies the correctness of the answer.

To check our solution, we substitute \(x = 5.5\) back into the original equation \(4(2x - 3) = 32\). The objective is to make sure the left and right sides are equal:

• Substitute: \(4(2 \times 5.5 - 3) = 32\)
• Simplify inside the parenthesis: \(2 \times 5.5 = 11\)
• Continue simplifying: \(11 - 3 = 8\)
• Distribute: \(4 \times 8 = 32\)

The left side equals the right side, which confirms that the solution \(x = 5.5\) is indeed correct. This final step of checking not only gives confidence in the solution but also helps in developing reliable problem-solving skills.