Math Problem Statement

2/x + 4 = x + 3

Solution

To solve the equation 2x+4=x+3\frac{2}{x} + 4 = x + 3, we need to isolate xx. Here's the step-by-step process:

Step 1: Subtract 3 from both sides

2x+43=x+33\frac{2}{x} + 4 - 3 = x + 3 - 3 This simplifies to: 2x+1=x\frac{2}{x} + 1 = x

Step 2: Subtract 1 from both sides

2x+11=x1\frac{2}{x} + 1 - 1 = x - 1 This simplifies to: 2x=x1\frac{2}{x} = x - 1

Step 3: Multiply both sides by xx to eliminate the denominator

x2x=(x1)xx \cdot \frac{2}{x} = (x - 1) \cdot x This simplifies to: 2=x(x1)2 = x(x - 1) Which expands to: 2=x2x2 = x^2 - x

Step 4: Rearrange the equation

x2x2=0x^2 - x - 2 = 0

Step 5: Factor the quadratic equation

x2x2=(x2)(x+1)=0x^2 - x - 2 = (x - 2)(x + 1) = 0

Step 6: Solve for xx

From (x2)(x+1)=0(x - 2)(x + 1) = 0, we get two solutions: x2=0orx+1=0x - 2 = 0 \quad \text{or} \quad x + 1 = 0 x=2orx=1x = 2 \quad \text{or} \quad x = -1

Step 7: Check for extraneous solutions

Since the original equation contains 2x\frac{2}{x}, x=0x = 0 is not a valid solution. However, x=2x = 2 and x=1x = -1 are valid.

Thus, the solutions are: x=2orx=1x = 2 \quad \text{or} \quad x = -1

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions:

  1. How do we solve equations with fractions and variables in the denominator?
  2. What is the method to factor quadratic equations?
  3. What is an extraneous solution, and why do we check for it?
  4. How do you complete the square to solve quadratic equations?
  5. How can we solve quadratic equations using the quadratic formula?

Tip: Always check for extraneous solutions when dealing with fractions in an equation to avoid invalid answers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Equations
Quadratic Equations

Formulas

Quadratic equation formula ax^2 + bx + c = 0
Factoring method

Theorems

Zero Product Property
Extraneous Solutions

Suitable Grade Level

Grades 9-11