Problem 82 - College Algebra with Modeling and Visualization

Question

Solve the equation. $$ \frac{3}{2 x+1}=-1 $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Eliminate the Fraction

To solve the equation \( \frac{3}{2x+1} = -1 \), first eliminate the fraction by multiplying both sides by \( 2x + 1 \). This results in the equation:\[ 3 = -1 (2x + 1). \]

Step 2: Simplify Both Sides

Distribute the \( -1 \) on the right side of the equation:\[ 3 = -2x - 1. \]

Step 3: Add or Subtract to Isolate the Term with the Variable

Add \( 1 \) to both sides of the equation to isolate the term with the variable:\[ 3 + 1 = -2x - 1 + 1. \]This simplifies to:\[ 4 = -2x. \]

Step 4: Solve for x

Divide both sides by \( -2 \) to solve for \( x \):\[ x = \frac{4}{-2} = -2. \]

Step 5: Verify the Solution

Substitute \( x = -2 \) back into the original equation to ensure it satisfies the equation:\[ \frac{3}{2(-2) + 1} = \frac{3}{-4 + 1} = \frac{3}{-3} = -1. \]This confirms the solution is correct.

Key Concepts

Step-by-Step Equation Solving

Step-by-step equation solving is a method where we break down the process into manageable stages. In this way, it's easier to understand and solve equations involving fractions or variables.

To solve the equation \( \frac{3}{2x+1} = -1 \), we first aim to eliminate the fraction. By doing so, we turn a seemingly complex equation into one that looks simpler.

• Multiply both sides by the denominator \(2x+1\), transforming the equation to \(3 = -1(2x+1)\).
• Expand the terms on the right side to eliminate the parentheses.

Move step-by-step, ensuring each transformation simplifies or isolates the variable further. This approach minimizes mistakes and leads us logically towards the solution.

Algebraic Manipulation

Algebraic manipulation involves operations such as addition, subtraction, multiplication, or division to rearrange and simplify equations. In our example, each step carefully applies these operations to maintain the equality of both sides.

Start by addressing the equation from the step where we eliminated the fraction:

• We have \(3 = -2x - 1\) after simplification.
• Add \(1\) to both sides: \(3 + 1 = -2x - 1 + 1\). This results in \(4 = -2x\).
• Now, divide both sides by \(-2\) to solve for \(x\). This gives \(x = \frac{4}{-2} = -2\).

In this example, manipulating terms step-by-step ensures each process smoothly transitions to the next, ultimately solving for the unknown. Each manipulation sees us getting closer to isolating \(x\) and finding its value.

Equation Verification

Equation verification is essential to confirm that our solution satisfies the original equation. This final step checks our work by revisiting the starting equation with the solution substituted for any variables.

For our equation, substitute \(x = -2\) back into \(\frac{3}{2x+1} = -1\):

• Replace \(x\) in the denominator: \(2(-2) + 1 = -4 + 1 = -3\).
• The left side becomes: \(\frac{3}{-3} = -1\).

Since both sides of the equation match, \(-1 = -1\), the solution is verified as correct. Verifying solutions not only ensures accuracy but reinforces understanding of how our derived answer fits within the problem's context.