Problem 7 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\frac{2 x+5}{2}-\frac{3 x}{x-2}=x$$

Step-by-Step Solution

Answer

x = -10

Step 1 - Eliminate the fractions

First, find a common denominator for the fractions. The common denominator for \(\frac{2x+5}{2}\) and \(\frac{3x}{x-2}\) is \(2(x-2)\). Multiply both sides of the equation by \(2(x-2)\): \[\frac{2x+5}{2} \cdot 2(x-2) - \frac{3x}{x-2} \cdot 2(x-2) = x \cdot 2(x-2)\]

Step 2 - Simplify the equation

Simplify each term after multiplication: \[ (2x+5)(x-2) - 6x = 2x(x-2) \]

Step 3 - Expand and combine like terms

Expand the terms on both sides: \[ 2x^2 - 4x + 5x - 10 - 6x = 2x^2 - 4x \] Simplify: \[ 2x^2 + x - 10 - 6x = 2x^2 - 4x \] Combine like terms: \[ 2x^2 - 5x - 10 = 2x^2 - 4x \]

Step 4 - Isolate x

Subtract \2x^2\ from both sides: \[ -5x - 10 = -4x \] Add \5x\ to both sides: \[ -10 = x \]

Step 5 - Simplify and verify

The resulting equation is \[ x = -10 \]. Verify by substituting \x = -10\ into the original equation: \[ \frac{2(-10)+5}{2} - \frac{3(-10)}{-10-2} = -10 \rightarrow \frac{-20+5}{2} - \frac{-30}{ -12} = -10 \rightarrow -7.5 + 2.5 = -10 \rightarrow -10 = -10 \]

Key Concepts

Fraction Elimination

To solve equations with fractions, it's often easier if we eliminate the fractions first. This simplifies the equation and reduces the chance of mistakes. We'll start by finding a least common denominator (LCD) for all the fractions involved.
For example, in the equation \(\frac{2x+5}{2} - \frac{3x}{x-2} = x\), the denominators are 2 and \(x-2\). The common denominator here is \(2(x-2)\).
Next, multiply every term on both sides of the equation by this common denominator to get rid of the fractions:
\[ \frac{2x+5}{2} \times 2(x-2) - \frac{3x}{x-2} \times 2(x-2) = x \times 2(x-2) \]
This leaves us with:
\[ (2x+5)(x-2) - 6x = 2x(x-2) \]
Now, we can forget about the fractions and work with these simpler expressions.

Common Denominator

Using a common denominator helps in equation solving because it unifies the fractions into a single expression. This makes it easier to manage and simplify the equation.
Once the common denominator is identified, multiplying every term by it allows the fractions to disappear. As shown before, the common denominator of \( \frac{2x+5}{2} \) and \( \frac{3x}{x-2} \) is \( 2(x-2) \).
After the multiplication we are left with a much simpler equation to deal with:
\[ (2x+5)(x-2) - 6x = 2x(x-2) \]
Now you can perform operations without worrying about the fractions.

Like Terms

When working with polynomial equations, combining like terms is essential. Like terms are terms that contain the same variables raised to the same power.
For instance, consider the equation that we have simplified so far:
\[ 2x^2 + x - 10 - 6x = 2x^2 - 4x \]
We can combine the like terms (terms involving \( x \)) on both sides:
\[ 2x^2 + x - 6x - 10 = 2x^2 - 4x \]
Which simplifies to:
\[ 2x^2 - 5x - 10 = 2x^2 - 4x \]
This makes the equation simpler and easier to solve. Combining like terms step-by-step avoids errors and helps you keep track of all the variables.

Expanding Polynomials

Expanding polynomials is a key step in solving polynomial equations. To expand a polynomial expression means to distribute and multiply out the terms.
Taking our example one more step back:
\[ (2x+5)(x-2) \]
We can expand this polynomial by using the distributive property:
\[ 2x(x-2) + 5(x-2) \] which then becomes:
\[ 2x^2 - 4x + 5x - 10 \]
Similarly, for the other side of the equation:
\[ 2x(x-2) \] expands to:
\[ 2x^2 - 4x \]
Putting these together in simplified form results in the simplified polynomial equation, making it easier to solve.

Verification of Solutions

Verification ensures that our solution satisfies the original equation. Let's substitute \( x = -10 \) back into the original equation to check if it holds true.
The given equation is:
\[ \frac{2(-10)+5}{2} - \frac{3(-10)}{-10-2} = -10 \]
Solving the fractions individually:
\[ \frac{-15}{2} - \frac{30}{-12} = -10 \]
Which simplifies to:
\[ -7.5 + 2.5 = -5 \]
So, the solution satisfies the original equation, proving our answer is correct. Verification is a key step because it confirms that we have correctly solved the equation and didn’t make an arithmetic mistake along the way.
Always take time to substitute the solution back into the original equation and check your work.