Problem 27 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solution of the equation is \(x = -1, -\frac{2}{3}\).

Step 1: Simplify the Equation

Firstly, multiply both sides of the equation by \((x+1)(x-1)\) to get rid of the fractions. This results in \(-3x(x-1) = -2(x+1)\).

Step 2: Expand Both Sides

Expanding both sides of the equation gives \(-3x^2 + 3x = -2x -2\).

Step 3: Rearrange the Equation

Next, rearrange the equation to have all terms on one side and zero on the other side. This yields \(3x^2 + 5x + 2 = 0\).

Step 4: Solve the Quadratic Equation

This is a simple quadratic equation in the form \(ax^2 + bx + c = 0\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve this equation. Substituting \(a = 3\), \(b = 5\), and \(c = 2\) yields two possible solutions: \(x = \frac{-5 + \sqrt{25 - 24}}{6} = -1\), and \(x = \frac{-5 - \sqrt{25 -24}}{6} = -\frac{2}{3}\).

Key Concepts

Equation Solving

Solving an equation is like solving a puzzle. You are searching for the value of the variable that makes the equation true. It often involves a series of steps like simplifying, rearranging, and solving sub-equations.
In our equation, \[ \frac{-3x}{x+1} = \frac{-2}{x-1} \]
we began by eliminating fractions. We did this by multiplying both sides by the denominators, creating a new polynomial equation without fractions.

• Handle each term carefully while multiplying.
• Ensure all like terms are combined properly.

By making the equation devoid of fractions, it becomes easier and straightforward to solve for the variable.

Rational Equations

Rational equations are equations containing fractions with variables in the denominators. Solving these requires removing the fractions first, often by finding a common denominator.
By multiplying both sides of the rational equation \[ \frac{-3x}{x+1} = \frac{-2}{x-1}, \]
by \((x+1)(x-1)\), we effectively got rid of the denominators. This step is crucial as it reduces a rational equation to a simpler polynomial form.
Remember:

• Always check for any restrictions on the variable to avoid denominators being zero.
• Be careful when expanding and rearranging.

This process of moving from a rational to a polynomial equation is key to simplifying the process of finding solutions.

Quadratic Formula

The quadratic formula is a reliable tool for solving quadratic equations, which have the general form \( ax^2 + bx + c = 0\).
For our equation, \( 3x^2 + 5x + 2 = 0\), we identified \(a = 3\), \(b = 5\), and \(c = 2\). Plugging these into the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \)we calculated the possible values for \(x\).
This formula:

• Gives potential solutions for any quadratic equation.
• Helps determine if solutions are real or complex based on the discriminant \(b^2 - 4ac\).

In this case, we found two solutions, \(x = -1\) and \(x = - \frac{2}{3}\), which reflect the points where the quadratic crosses the x-axis.