Problem 8 - Algebra A Combined Function

Question

If the difference of a number and four is doubled, the result is \(\frac{1}{4}\) less than the number. Find the number.

Step-by-Step Solution

Answer

The number is \( \frac{31}{4} \).

Step 1: Set up the Equation

Let's define the number as \( x \). The problem states that the difference between the number and four is doubled. This can be expressed as \( 2(x - 4) \). The problem also states that this result is \( \frac{1}{4} \) less than the number \( x \). Therefore, we can set up the equation: \[ 2(x - 4) = x - \frac{1}{4}. \]

Step 2: Simplify the Equation

First, simplify the left side of the equation: \( 2(x - 4) = 2x - 8 \). Substitute this back into the equation, we get \[ 2x - 8 = x - \frac{1}{4}. \]

Step 3: Move Terms Involving x to One Side

Subtract \( x \) from both sides to get all terms involving \( x \) on one side of the equation: \[ 2x - x - 8 = - \frac{1}{4}. \] This simplifies to \( x - 8 = -\frac{1}{4} \).

Step 4: Isolate x

Add 8 to both sides to solve for \( x \): \[ x - 8 + 8 = - \frac{1}{4} + 8. \] Simplifying the right side gives: \( x = \frac{31}{4} \).

Step 5: Verify the Solution

Calculate to verify: If \( x = \frac{31}{4} \), the difference \( x - 4 \) is \( \frac{31}{4} - \frac{16}{4} = \frac{15}{4} \). Doubling it gives \( \frac{30}{4} \). The expression for \( x \) that is \( \frac{1}{4} \) less than the number is \( \frac{31}{4} - \frac{1}{4} = \frac{30}{4} \), which confirms our solution.

Key Concepts

Equation Setup

In any algebra problem solving, setting up the equation accurately is crucial. In this problem, we assume the unknown number as \( x \). The exercise gives a condition: the difference between this number and 4 is first calculated. We represent this as \( x - 4 \).Next, the difference is doubled. Applying basic algebra, we can express it as \( 2(x - 4) \). The problem states this result is \( \frac{1}{4} \) less than the original number. Therefore, we write this relationship with our unknown number \( x \), leading to the equation:

• \( 2(x - 4) = x - \frac{1}{4} \)

This equation setup is the first step and begins our journey into solving the problem mathematically.

Simplification

Once our equation is properly set up, the next step is simplification. Simplification makes complex algebraic expressions easier to work with by combining like terms and reducing fractions if needed. For our given equation, \( 2(x - 4) = x - \frac{1}{4} \), simplification allows us to break this down:

• First, expand \( 2(x - 4) \) to become \( 2x - 8 \)
• Substitute this simplification back to get \( 2x - 8 = x - \frac{1}{4} \)

These small steps are essential to move forward. Always look for possibilities to combine terms or open brackets to simplify your work. This helps in creating a clearer path towards solving for the unknown.

Isolation of Variable

Isolating the variable is a key concept in solving equations, as it leads us directly to the solution. Think of this as getting \( x \) all by itself on one side of the equation.Starting with our simplified equation \( 2x - 8 = x - \frac{1}{4} \), our goal is to gather all \( x \) terms on one side:

• Subtract \( x \) from both sides: \( 2x - x - 8 = -\frac{1}{4} \)
• Which simplifies to \( x - 8 = -\frac{1}{4} \)

Now, finally, solve for \( x \):

• Add 8 to both sides to isolate \( x \): \( x = -\frac{1}{4} + 8 \)
• Simplify the right side to find \( x = \frac{31}{4} \)

Knowing how to manipulate and rearrange the equation is an invaluable part of algebra problem solving.

Verification of Solution

Verifying your solution is the final and crucial step. It ensures that the solution obtained is indeed correct. For this exercise, let’s check if \( x = \frac{31}{4} \) satisfies the original conditions.Start by computing the left-side condition:

• \( x - 4 = \frac{31}{4} - \frac{16}{4} = \frac{15}{4} \)
• Doubling it gives \( \frac{30}{4} \)

Now, check the right-side condition where it's \( \frac{1}{4} \) less than \( x \):

• \( x - \frac{1}{4} = \frac{31}{4} - \frac{1}{4} = \frac{30}{4} \)

Since both expressions match, the verification confirms our solution is correct. Always verify after solving—this step often catches tiny errors and reinforces confidence in your answer.