Problem 24 - Prealgebra

Question

Solve each equation. $$0.11 x+0.12(x+4000)=940$$

Step-by-Step Solution

Answer

The solution is \(x = 2000\).

Step 1: Distribute 0.12

To eliminate the parentheses, distribute 0.12 to both terms inside the parentheses in the equation: \(0.11x + 0.12(x + 4000) = 940\)This results in:\(0.11x + 0.12x + 480 = 940\) since \(0.12 \times 4000 = 480\).

Step 2: Combine like terms

Combine the \(x\)-terms on the left side of the equation:\(0.11x + 0.12x + 480 = 940\)This simplifies to:\(0.23x + 480 = 940\).

Step 3: Isolate the variable term

Subtract 480 from both sides to isolate the term with \(x\):\(0.23x + 480 - 480 = 940 - 480\)This results in:\(0.23x = 460\).

Step 4: Solve for x

Divide both sides by 0.23 to solve for \(x\):\(x = \frac{460}{0.23}\).Perform the division to find:\(x = 2000\).

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics that involves finding the value of the variable that makes the equation true. An equation is essentially a mathematical statement asserting that two expressions are equal. In the equation given, we have:\[ \begin{align*} 0.11x + 0.12(x + 4000) &= 940 \end{align*} \]

• Our goal is to find the value of \(x\) such that both sides of the equation are equal.
• This often involves several steps such as distributing, combining like terms, isolating the variable, and solving for the unknown.
• You perform the operations needed to simplify and rearrange the equation until the variable \(x\) is isolated on one side.

Mastering the art of solving equations opens up the world to more complex algebraic expressions and is a critical skill in higher mathematics.

Distributive Property

The distributive property is a valuable tool in mathematics that allows us to manipulate expressions in a simpler form. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.In our given equation:\[ \begin{align*} 0.12(x + 4000) \end{align*} \]The distributive property is applied by multiplying 0.12 with each term inside the parentheses:

• This results in \(0.12x + 480\) because \(0.12 \times 4000 = 480\).
• We have now eliminated the parentheses and made our equation more straightforward.

Using this property effectively helps simplify complex expressions and solve equations more efficiently. Understanding and applying the distributive property is a stepping stone to more complex algebraic concepts.

Combining Like Terms

After applying the distributive property, the next step often involves combining like terms. Like terms are terms in an expression that have the same variable raised to the same power.In our equation:\[ \begin{align*} 0.11x + 0.12x + 480 &= 940 \end{align*} \]We combine the terms involving \(x\):

• \(0.11x\) and \(0.12x\) are like terms. Adding them together gives us \(0.23x\).
• This simplifies the equation to: \(0.23x + 480 = 940\).

Combining like terms is crucial for simplifying equations, thus making it easier to isolate the variable and solve the equation. It's a routine yet essential step in algebra.

Prealgebra Steps

Understanding prealgebra is a vital stage in developing algebraic thinking and problem-solving skills. The steps usually involve:

• Simplifying expressions using arithmetic operations.
• Identifying and applying properties such as distributive property.
• Combining and simplifying like terms efficiently.

Prealgebra sets the foundation for tackling more complex equations as shown in the example where each step builds upon the previous. First, we distribute, then combine like terms, and finally isolate the variable to solve for \(x\). These structured steps refine your ability to handle algebraic equations and bolster confidence in handling higher mathematical concepts.