Problem 81 - College Algebra with Modeling and Visualization

Question

Solve the equation. $$ \frac{4}{x+2}=-4 $$

Step-by-Step Solution

Answer

x = -3.

Step 1: Eliminate the Fraction

To eliminate the fraction, we will multiply both sides of the equation by the denominator, \(x+2\), to remove the fraction.\[4 = (-4)(x+2)\]

Step 2: Distribute the -4

Distribute the -4 on the right side of the equation:\[4 = -4x - 8\]

Step 3: Move Constants to One Side

Add 8 to both sides of the equation to move the constant term to one side:\[4 + 8 = -4x\]\[12 = -4x\]

Step 4: Solve for x

Divide both sides by -4 to solve for \(x\):\[\frac{12}{-4} = x\]\[x = -3\]

Key Concepts

Fraction Elimination

When you encounter an algebraic equation involving a fraction, your first step is often to eliminate the fraction to simplify the equation. This process is called fraction elimination. The basic idea is to multiply every term in the equation by the denominator of the fraction.

• This action removes the fraction, leaving you with a simpler, more manageable equation.
• You can only eliminate the fraction by using the same operations on both sides of the equation to maintain balance.

For example, if you have the equation \( \frac{4}{x+2} = -4 \), you need to multiply both sides by \( x+2 \). This step effectively cancels out the fraction, resulting in a new equation:\[4 = (-4)(x + 2)\] Now, your focus can shift to solving this linear equation without the complication of a fraction.

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to simplify expressions by multiplying a single term by each term inside a parenthesis. This property states that for any numbers or variables, \( a(b + c) = ab + ac \).
The key advantage of using the distributive property is that it helps you

• expand expressions,
• simplify equations,
• and make them easier to solve.

In the context of our example,
After eliminating the fraction, we have:\[ 4 = -4(x + 2) \]
Applying the distributive property, \(-4\) is multiplied by each term inside the parenthesis (\(x\) and \(2\)), which results in:\[ 4 = -4x - 8 \]Using this step, we break down complex equations into simpler parts, paving the way for solving the equation.

Solving Linear Equations

Once the fraction is eliminated and the distributive property is applied, the next task is to solve the resulting linear equation. A linear equation, like \( -4x - 8 = 4 \), is a polynomial equation of the first degree.
Start by moving constant terms to one side so that you isolate the variable term. In this case, add 8 to both sides:
\[ 4 + 8 = -4x \]
Which simplifies to:\[ 12 = -4x \]

• To isolate \(x\), divide both sides by \(-4\),
• resulting in \(x = \frac{12}{-4}\).

Simplifying the fraction gives:
\(x = -3\)
Thus, by carefully following steps like fraction elimination and applying properties such as distributive, you can systematically solve linear equations for clearly finding the value of variables.