Solve (y + 3)/(2y + 2) - (y + 1)/(2y - 2) + 3/(y ^ 2 - 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

$$\frac{y+3}{2y+2}-\frac{y+1}{2y-2}+\frac{3}{y^{2}-1}$$

Evaluate

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

Short Solution Steps

Factor $2y+2$. Factor $2y-2$.

$$\frac{y+3}{2\left(y+1\right)}-\frac{y+1}{2\left(y-1\right)}+\frac{3}{y^{2}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2\left(y+1\right)$ and $2\left(y-1\right)$ is $2\left(y-1\right)\left(y+1\right)$. Multiply $\frac{y+3}{2\left(y+1\right)}$ times $\frac{y-1}{y-1}$. Multiply $\frac{y+1}{2\left(y-1\right)}$ times $\frac{y+1}{y+1}$.

$$\frac{\left(y+3\right)\left(y-1\right)}{2\left(y-1\right)\left(y+1\right)}-\frac{\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Since $\frac{\left(y+3\right)\left(y-1\right)}{2\left(y-1\right)\left(y+1\right)}$ and $\frac{\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y+3\right)\left(y-1\right)-\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Do the multiplications in $\left(y+3\right)\left(y-1\right)-\left(y+1\right)\left(y+1\right)$.

$$\frac{y^{2}-y+3y-3-y^{2}-y-y-1}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Combine like terms in $y^{2}-y+3y-3-y^{2}-y-y-1$.

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

Factor $y^{2}-1$.

$$\frac{-4}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{\left(y-1\right)\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2\left(y-1\right)\left(y+1\right)$ and $\left(y-1\right)\left(y+1\right)$ is $2\left(y-1\right)\left(y+1\right)$. Multiply $\frac{3}{\left(y-1\right)\left(y+1\right)}$ times $\frac{2}{2}$.

$$\frac{-4}{2\left(y-1\right)\left(y+1\right)}+\frac{3\times 2}{2\left(y-1\right)\left(y+1\right)}$$

Since $\frac{-4}{2\left(y-1\right)\left(y+1\right)}$ and $\frac{3\times 2}{2\left(y-1\right)\left(y+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-4+3\times 2}{2\left(y-1\right)\left(y+1\right)}$$

Do the multiplications in $-4+3\times 2$.

$$\frac{-4+6}{2\left(y-1\right)\left(y+1\right)}$$

Do the calculations in $-4+6$.

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

Expand $2\left(y-1\right)\left(y+1\right)$.

$$\frac{2}{2y^{2}-2}$$

Expand

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

Short Solution Steps

Factor $2y+2$. Factor $2y-2$.

$$\frac{y+3}{2\left(y+1\right)}-\frac{y+1}{2\left(y-1\right)}+\frac{3}{y^{2}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2\left(y+1\right)$ and $2\left(y-1\right)$ is $2\left(y-1\right)\left(y+1\right)$. Multiply $\frac{y+3}{2\left(y+1\right)}$ times $\frac{y-1}{y-1}$. Multiply $\frac{y+1}{2\left(y-1\right)}$ times $\frac{y+1}{y+1}$.

$$\frac{\left(y+3\right)\left(y-1\right)}{2\left(y-1\right)\left(y+1\right)}-\frac{\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Since $\frac{\left(y+3\right)\left(y-1\right)}{2\left(y-1\right)\left(y+1\right)}$ and $\frac{\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y+3\right)\left(y-1\right)-\left(y+1\right)\left(y+1\right)}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Do the multiplications in $\left(y+3\right)\left(y-1\right)-\left(y+1\right)\left(y+1\right)$.

$$\frac{y^{2}-y+3y-3-y^{2}-y-y-1}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{y^{2}-1}$$

Combine like terms in $y^{2}-y+3y-3-y^{2}-y-y-1$.

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

Factor $y^{2}-1$.

$$\frac{-4}{2\left(y-1\right)\left(y+1\right)}+\frac{3}{\left(y-1\right)\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2\left(y-1\right)\left(y+1\right)$ and $\left(y-1\right)\left(y+1\right)$ is $2\left(y-1\right)\left(y+1\right)$. Multiply $\frac{3}{\left(y-1\right)\left(y+1\right)}$ times $\frac{2}{2}$.

$$\frac{-4}{2\left(y-1\right)\left(y+1\right)}+\frac{3\times 2}{2\left(y-1\right)\left(y+1\right)}$$

Since $\frac{-4}{2\left(y-1\right)\left(y+1\right)}$ and $\frac{3\times 2}{2\left(y-1\right)\left(y+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-4+3\times 2}{2\left(y-1\right)\left(y+1\right)}$$

Do the multiplications in $-4+3\times 2$.

$$\frac{-4+6}{2\left(y-1\right)\left(y+1\right)}$$

Do the calculations in $-4+6$.

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

Expand $2\left(y-1\right)\left(y+1\right)$.

$$\frac{2}{2y^{2}-2}$$