Solve (1 - x)/(1 + x) - (1 + x)/(4 - x) - (4x)/(1 - x ^ 2) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{1-x}{1+x}-\frac{1+x}{4-x}-\frac{4x}{1-x^{2}}$$

Evaluate

$\frac{11x^{2}-26x+3}{\left(4-x\right)\left(1-x^{2}\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $1+x$ and $4-x$ is $\left(x+1\right)\left(-x+4\right)$. Multiply $\frac{1-x}{1+x}$ times $\frac{-x+4}{-x+4}$. Multiply $\frac{1+x}{4-x}$ times $\frac{x+1}{x+1}$.

$$\frac{\left(1-x\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Since $\frac{\left(1-x\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}$ and $\frac{\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(1-x\right)\left(-x+4\right)-\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Do the multiplications in $\left(1-x\right)\left(-x+4\right)-\left(1+x\right)\left(x+1\right)$.

$$\frac{4-x+x^{2}-4x-x-1-x^{2}-x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Combine like terms in $4-x+x^{2}-4x-x-1-x^{2}-x$.

$$\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Factor $1-x^{2}$.

$$\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{\left(x-1\right)\left(-x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(-x+4\right)$ and $\left(x-1\right)\left(-x-1\right)$ is $\left(x-4\right)\left(x-1\right)\left(x+1\right)$. Multiply $\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}$ times $\frac{-\left(x-1\right)}{-\left(x-1\right)}$. Multiply $\frac{4x}{\left(x-1\right)\left(-x-1\right)}$ times $\frac{-\left(x-4\right)}{-\left(x-4\right)}$.

$$\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}-\frac{4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Since $\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$ and $\frac{4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)-4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in $\left(3-7x\right)\left(-1\right)\left(x-1\right)-4x\left(-1\right)\left(x-4\right)$.

$$\frac{-3x+3+7x^{2}-7x+4x^{2}-16x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Combine like terms in $-3x+3+7x^{2}-7x+4x^{2}-16x$.

$$\frac{-26x+3+11x^{2}}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Expand $\left(x-4\right)\left(x-1\right)\left(x+1\right)$.

$$\frac{-26x+3+11x^{2}}{x^{3}-4x^{2}-x+4}$$

Expand

$\frac{11x^{2}-26x+3}{\left(4-x\right)\left(1-x^{2}\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $1+x$ and $4-x$ is $\left(x+1\right)\left(-x+4\right)$. Multiply $\frac{1-x}{1+x}$ times $\frac{-x+4}{-x+4}$. Multiply $\frac{1+x}{4-x}$ times $\frac{x+1}{x+1}$.

$$\frac{\left(1-x\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Since $\frac{\left(1-x\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}$ and $\frac{\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(1-x\right)\left(-x+4\right)-\left(1+x\right)\left(x+1\right)}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Do the multiplications in $\left(1-x\right)\left(-x+4\right)-\left(1+x\right)\left(x+1\right)$.

$$\frac{4-x+x^{2}-4x-x-1-x^{2}-x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Combine like terms in $4-x+x^{2}-4x-x-1-x^{2}-x$.

$$\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{1-x^{2}}$$

Factor $1-x^{2}$.

$$\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}-\frac{4x}{\left(x-1\right)\left(-x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(-x+4\right)$ and $\left(x-1\right)\left(-x-1\right)$ is $\left(x-4\right)\left(x-1\right)\left(x+1\right)$. Multiply $\frac{3-7x}{\left(x+1\right)\left(-x+4\right)}$ times $\frac{-\left(x-1\right)}{-\left(x-1\right)}$. Multiply $\frac{4x}{\left(x-1\right)\left(-x-1\right)}$ times $\frac{-\left(x-4\right)}{-\left(x-4\right)}$.

$$\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}-\frac{4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Since $\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$ and $\frac{4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3-7x\right)\left(-1\right)\left(x-1\right)-4x\left(-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in $\left(3-7x\right)\left(-1\right)\left(x-1\right)-4x\left(-1\right)\left(x-4\right)$.

$$\frac{-3x+3+7x^{2}-7x+4x^{2}-16x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Combine like terms in $-3x+3+7x^{2}-7x+4x^{2}-16x$.

$$\frac{-26x+3+11x^{2}}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Expand $\left(x-4\right)\left(x-1\right)\left(x+1\right)$.

$$\frac{-26x+3+11x^{2}}{x^{3}-4x^{2}-x+4}$$