Problem 4 - Prealgebra and Introductory Algebra

Question

$$\text { If } x \neq-2 \text { and } x \neq 0, \text { then } \frac{x}{x+2}+\frac{3}{x+2}=\frac{x+3}{x+2}=\frac{3}{2}$$

Step-by-Step Solution

Answer

However, from the original conditions specified in the problem, x cannot be 0. Therefore, the given equation has no solution.

Step 1: Simplify the equation

Since the expression on the left simplifies to the right (the two fractions become one), the equation to solve becomes: \( \frac{x+3}{x+2}=\frac{3}{2} \)

Step 2: Cross-multiply

Cross-multiply the terms to get rid of the fraction: \( (x + 3) * 2 = (x + 2) * 3 \)

Step 3: Distribute

Distribute the values on both sides: \( 2x + 6 = 3x + 6 \)

Step 4: Solve for 'x'

Next, subtract \(2x\) from both sides to isolate the variable on one side: \( 6 = x + 6\)

Step 5: Solve for 'x'

Finally, subtract 6 from both sides to completely isolate 'x': \( x = 0\)

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. When you have fractions with the same denominator, like \( \frac{x}{x+2} \) and \( \frac{3}{x+2} \), you can directly add them by summing up the numerators, as they are part of the same fractional part.
This is because the denomination represents how many parts the whole is divided into, enabling the numerator to simply combine.

• In general, if you have \( \frac{a}{c} + \frac{b}{c} \), this can be simplified to \( \frac{a+b}{c} \).
• It’s important to understand that fraction simplification can significantly ease solving the equations, making the operations manageable.

This simplification reduces the fractions' complexity, allowing further operations to be simpler and more straightforward.

Equation Solving

Equation solving involves finding the value of the variable that makes the equality true. In the given problem, once the fractions are simplified, the resulting equation is \( \frac{x+3}{x+2} = \frac{3}{2} \).
Solving this type of fraction-based equation requires techniques like cross-multiplication to clear out the fractions.

• First, ensure that the equation is accurately simplified.
• Apply cross-multiplication (a technique that involves multiplying across the fractions).

Once the fractions are removed, you are left with a simpler algebraic equation that you can solve for the unknown variable to find its value.

Cross-Multiplication

Cross-multiplication is a method used to solve equations that involve fractions. When you have an equation like \( \frac{a}{b} = \frac{c}{d} \), you multiply across the equation.
This means multiplying \( a \) by \( d \), and \( b \) by \( c \), setting them equal to each other:

• The equation \( \frac{x+3}{x+2} = \frac{3}{2} \) becomes \((x + 3) \times 2 = (x + 2) \times 3 \).
• This operation eliminates the fraction part, simplifying the problem into a form that is easier to handle.

Cross-multiplication helps in balancing the equation by avoiding fractions, reducing the likelihood of step errors, thus making it easier to manipulate and solve for the variable.

Isolation of Variable

Isolation of a variable means rearranging an equation so that the unknown variable stands alone on one side of the equation. This technique is essential for solving equations.
After applying cross-multiplication and simplifying, we get \( 2x + 6 = 3x + 6 \).

• To isolate \( x \), subtract \( 2x \) from both sides, yielding \( 6 = x + 6 \).
• Further simplify by subtracting 6 from both sides to fully isolate \( x \), giving \( x = 0 \).

The goal of variable isolation is to reach a stage where the unknown variable equals a constant or an expression that allows computation of its value.