Math Problem Statement

2/x+1>x-1/x+1

Solution

To solve the inequality:

2x+1>x1x+1\frac{2}{x+1} > x - \frac{1}{x+1}

we first need to combine like terms by getting everything over a common denominator.

Step 1: Combine like terms

The common denominator is x+1x + 1, so we can rewrite the inequality as:

2x+1>x(x+1)x+11x+1\frac{2}{x+1} > \frac{x(x+1)}{x+1} - \frac{1}{x+1}

Simplify the right-hand side:

2x+1>x2+x1x+1\frac{2}{x+1} > \frac{x^2 + x - 1}{x+1}

Now, we have:

2x+1>x2+x1x+1\frac{2}{x+1} > \frac{x^2 + x - 1}{x+1}

Step 2: Cross-multiply and solve

Since the denominators are the same, we can compare the numerators directly:

2>x2+x12 > x^2 + x - 1

Rearrange to form a quadratic inequality:

x2+x3<0x^2 + x - 3 < 0

Step

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Math Problem Analysis

Mathematical Concepts

Inequalities
Rational Expressions
Quadratic Inequalities

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12