Solve (4y)/(y ^ 2 - 1) - (y + 1)/(y - 1) | 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

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

Evaluate

$\frac{1-y}{y+1}$

Short Solution Steps

Factor $y^{2}-1$.

$$\frac{4y}{\left(y-1\right)\left(y+1\right)}-\frac{y+1}{y-1}$$

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

$$\frac{4y}{\left(y-1\right)\left(y+1\right)}-\frac{\left(y+1\right)\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

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

$$\frac{4y-\left(y+1\right)\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

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

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

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

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

Factor the expressions that are not already factored in $\frac{2y-y^{2}-1}{\left(y-1\right)\left(y+1\right)}$.

$$\frac{\left(y-1\right)\left(-y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

Extract the negative sign in $1-y$.

$$\frac{-\left(y-1\right)\left(y-1\right)}{\left(y-1\right)\left(y+1\right)}$$

Cancel out $y-1$ in both numerator and denominator.

$$\frac{-\left(y-1\right)}{y+1}$$

To find the opposite of $y-1$, find the opposite of each term.

$$\frac{-y+1}{y+1}$$

Expand

$-\frac{y-1}{y+1}$

Short Solution Steps

Factor $y^{2}-1$.

$$\frac{4y}{\left(y-1\right)\left(y+1\right)}-\frac{y+1}{y-1}$$

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

$$\frac{4y}{\left(y-1\right)\left(y+1\right)}-\frac{\left(y+1\right)\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

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

$$\frac{4y-\left(y+1\right)\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

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

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

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

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

Factor the expressions that are not already factored in $\frac{2y-y^{2}-1}{\left(y-1\right)\left(y+1\right)}$.

$$\frac{\left(y-1\right)\left(-y+1\right)}{\left(y-1\right)\left(y+1\right)}$$

Extract the negative sign in $1-y$.

$$\frac{-\left(y-1\right)\left(y-1\right)}{\left(y-1\right)\left(y+1\right)}$$

Cancel out $y-1$ in both numerator and denominator.

$$\frac{-\left(y-1\right)}{y+1}$$

To find the opposite of $y-1$, find the opposite of each term.

$$\frac{-y+1}{y+1}$$