Problem 43 - Prealgebra

Question

Solve the equation. \(-0.8(0.3 x+0.4)=-11.3\)

Step-by-Step Solution

Answer

\(x = 45.75\)

Step 1: Distribute -0.8

Multiply -0.8 by each term inside the parentheses. This gives: \[-0.8 \times (0.3x) + (-0.8) \times 0.4 = -11.3\] This simplifies to: \[-0.24x - 0.32 = -11.3\]

Step 2: Move Constant to the Other Side

Add 0.32 to both sides of the equation to isolate the term with \(x\) on one side:\[-0.24x = -11.3 + 0.32\]This simplifies to:\[-0.24x = -10.98\]

Step 3: Solve for x

Divide both sides of the equation by -0.24 to solve for \(x\):\[x = \frac{-10.98}{-0.24}\]Calculating the division gives:\[x = 45.75\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps distribute a term across terms inside a parenthesis. This means multiplying the term outside the parentheses by each term inside. It's like unpacking a gift and ensuring every piece gets its part.Here's how it works for the equation \[-0.8(0.3x + 0.4) = -11.3\]:

• Multiply \(-0.8\) by \(0.3x\) to get \(-0.24x\).
• Multiply \(-0.8\) by \(0.4\) to get \(-0.32\).

After applying the distributive property, the equation simplifies to:\[-0.24x - 0.32 = -11.3\]Now, it's easier to work with because the equation is free from parentheses, paving the way for the next steps in solving the equation.

Isolation of Variable

Once you've used the distributive property, the next step is to isolate the variable, in this case, \(x\). This process involves moving terms around so that the target variable is alone on one side of the equation. For example, with the equation \[-0.24x - 0.32 = -11.3\], we want \(x\) to be by itself. We do this by using addition or subtraction.How to isolate:

• Add \(0.32\) to both sides to remove the constant from the left: \[-0.24x - 0.32 + 0.32 = -11.3 + 0.32\]
• This simplifies to \[-0.24x = -10.98\].

Now, \(x\) is tied to the coefficient \(-0.24\), but we have successfully cleared out any other constants. This helps in preparing the equation for the final solution step.

Division in Equations

The last step involves isolation through division. Here, we focus on getting \(x\) entirely by itself by removing the coefficient attached to it through division. Looking at the reduced equation:\[-0.24x = -10.98\],we can eliminate the \(-0.24\) by dividing both sides of the equation by this coefficient. The division process includes:

• Divide both sides by \(-0.24\): \[x = \frac{-10.98}{-0.24}\]
• Perform the calculation: \[x = 45.75\]

The division effectively completes the isolation of \(x\), providing us with the solution to the equation. Through division, you've turned an equation into a simple statement of equality that gives the value of \(x\). This step emphasizes the balance of equations—whatever you do to one side, you must do to the other to keep the equation true.