Problem 36 - College Algebra

Question

For the following exercises, solve for the variable. $$ 4(3-1) x=4 $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(\frac{1}{2}\).

Step 1: Simplify the expression inside the parentheses

Begin by simplifying the expression inside the parentheses. The expression is \(3 - 1\), which simplifies to \(2\).

Step 2: Simplify the left side

Substitute the simplified expression into the equation. The equation becomes: \[4 \times 2 \times x = 4\] This simplifies to: \[8x = 4\]

Step 3: Solve for \(x\)

Now, solve for \(x\) by dividing both sides of the equation by \(8\):\[x = \frac{4}{8}\]Simplify the fraction to get \(x = \frac{1}{2}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, making complex problems easier to solve. Imagine you're simplifying a shopping list by counting items instead of remembering details. In our example, the expression inside the parentheses, \(3 - 1\), is simplified to \(2\).
This step reduces complexity and we can focus on solving form. Begin by identifying parts of the expression that can be combined or simplified.
Substituting this simplified value back, transforms the original problem into the next manageable piece.
Think of simplification as tidying up an equation so you’re working with as simple a version as possible.

• Evaluate expressions inside parentheses first.
• Perform any mathematical operations like addition or subtraction.

By understanding and applying these steps, solving expressions becomes much more manageable.

Algebraic Equations

Algebraic equations are important in mathematics because they allow us to find the value of unknown variables, like \(x\) in our example. The key relationship in algebraic equations is equality, represented by the \(=\) sign.
In the exercise, once the equation is simplified to \(8x = 4\), it becomes an algebraic equation where one side is a known value and the other side contains the variable, \(x\).
To solve an algebraic equation, you often perform operations to isolate the variable. This process ensures that the equation stays balanced.
Some steps to keep in mind are:

• Use inverse operations like division or multiplication to isolate the variable.
• Maintain equality by applying the same operation to both sides of the equation.

Understanding these principles forms the basis of solving more complex algebraic problems in the future.

Basic Algebra Concepts

Basic algebra concepts are the building blocks of solving equations and understanding algebra. These concepts include variables, operations, and the rules of equality.
A variable, like \(x\), represents an unknown value we are solving for.
Operations such as multiplication, division, addition, or subtraction are used to manipulate equations. The rules of equality ensure that the two sides of an equation are balanced.
From our example, the goal is to isolate the variable, which requires a firm grasp of these principles.
Here are a few basic concepts to remember:

• Variables represent unknown numbers or values.
• Perform operations carefully to both sides to keep equations balanced.
• Simplify expressions wherever possible to make solving easier.

Understanding these basic concepts will make it easier to tackle a variety of mathematical problems.