Problem 67 - Contemporary Precalculus

Question

Solve the equation and check your answers. $$\frac{2 x-7}{x+4}=\frac{5}{x+4}-2$$

Step-by-Step Solution

Answer

Question: Solve the following linear rational equation and verify your solution: $$\frac{2x-7}{x+4}=\frac{5}{x+4}-2$$ Answer: The solution to the given equation is \(x=1\). This has been confirmed by checking that the simplified expressions indicate equality when the value of \(x\) is substituted back into the original equation.

Step 1: Eliminate the denominators

Since both fractions have the same denominator (\(x+4\)), we can multiply both sides of the equation by \((x+4)\) to get rid of the denominator: $$(x+4)\left(\frac{2x-7}{x+4}\right)=(x+4)\left(\frac{5}{x+4}-2\right)$$ This simplifies to: $$(2x-7) = (5-2(x+4))$$

Step 2: Expand and further simplify the equation

Distribute the -2 on the right side of the equation: $$(2x-7) = (5-2x-8)$$ Now, simplify the right side by combining the constants: $$(2x-7) = (-2x-3)$$

Step 3: Solve for x

To solve for \(x\), let's first move the terms containing \(x\) to one side of the equation by adding \(2x\) to both sides: $$4x-7 = -3$$ Now, add \(7\) to both sides to isolate \(x\): $$4x = 4$$ Finally, divide both sides by \(4\) to get the value of \(x\): $$x = 1$$

Step 4: Check the answer

Now, let's check if the value of \(x\) we found satisfies the original equation: $$\frac{2(1)-7}{1+4}=\frac{5}{1+4}-2$$ We get: $$\frac{-5}{5}=\frac{5}{5}-2$$ Simplifying both fractions: $$-1 = 1-2$$ As the simplified expressions are equal, we can confirm that the solution \(x=1\) is correct.

Key Concepts

Eliminate Denominators

The first step in handling equations with fractions is often to eliminate the denominators. Why do we do this? By removing denominators, you transform a rational equation into a simpler, often linear equation, which is easier to solve.

When faced with a rational equation like \(\frac{2x-7}{x+4}=\frac{5}{x+4}-2\), observe if the fractions share a common denominator. If they do, you can multiply each term by that common denominator to eliminate the fractional parts altogether. This is what we did in the exercise, multiplying every term by \(x+4\) leading to the simpler equation \(2x - 7 = 5 - 2(x + 4)\).

Remember that when there are multiple terms with denominators, you need to apply the least common denominator (LCD) for all of them, which may be a multiple of individual denominators.

Simplifying Equations

Once you've eliminated the denominators, the next step is to simplify the equation. This involves expanding any parentheses and combining like terms. You simplify in order to bring the equation into a form where the variable you want to solve for, \(x\), is easier to isolate.

For the given exercise, after eliminating the denominators, expanding the parentheses (\(5 - 2(x + 4)\)) gave us \(2x - 7 = -2x - 3\). Then we combined like terms by adding \(2x\) to both sides, resulting in a more manageable equation, \(4x - 7 = -3\). Simplification is critical because it lays the groundwork for isolating the variable and finding the solution to the equation.

Checking Solutions

The final and crucial step is to check your solutions. This step ensures that the solution you've determined actually satisfies the original equation, especially since in rational equations, there can be false solutions introduced by the steps of solving.

To check the solution \(x = 1\), we substitute it back into the original equation and simplify: \(\frac{2(1)-7}{1+4}=\frac{5}{1+4}-2\) simplifies to \(\frac{-5}{5}=\frac{5}{5}-2\), and further to \( -1 = 1 - 2\). Since both sides of the equation are equal after substitution, our solution is correct. Checking is important to avoid accepting 'extraneous solutions' that don't actually satisfy the equation.