Solve (x+2)/(3x-1)+(x+1)/(3-2x)+(4(x)^(2)+6x+3)/(6(x)^(2)-11x+3) | 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

$$\frac{ x+2 }{ 3x-1 } + \frac{ x+1 }{ 3-2x } + \frac{ 4 { x }^{ 2 } +6x+3 }{ 6 { x }^{ 2 } -11x+3 }$$

Evaluate

$\frac{x+2}{2x-3}$

Short Solution Steps

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

$$\frac{\left(x+2\right)\left(-2x+3\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Do the multiplications in $\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)$.

$$\frac{-2x^{2}+3x-4x+6+3x^{2}-x+3x-1}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Combine like terms in $-2x^{2}+3x-4x+6+3x^{2}-x+3x-1$.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Factor $6x^{2}-11x+3$.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(3x-1\right)\left(-2x+3\right)$ and $\left(2x-3\right)\left(3x-1\right)$ is $\left(2x-3\right)\left(3x-1\right)$. Multiply $\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}$ times $\frac{-1}{-1}$.

$$\frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)}+\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Since $\frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)}$ and $\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-\left(x^{2}+x+5\right)+4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Do the multiplications in $-\left(x^{2}+x+5\right)+4x^{2}+6x+3$.

$$\frac{-x^{2}-x-5+4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Combine like terms in $-x^{2}-x-5+4x^{2}+6x+3$.

$$\frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}$$

Factor the expressions that are not already factored in $\frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}$.

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

Cancel out $3x-1$ in both numerator and denominator.

$$\frac{x+2}{2x-3}$$

Expand

$\frac{x+2}{2x-3}$

Short Solution Steps

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

$$\frac{\left(x+2\right)\left(-2x+3\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Do the multiplications in $\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)$.

$$\frac{-2x^{2}+3x-4x+6+3x^{2}-x+3x-1}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Combine like terms in $-2x^{2}+3x-4x+6+3x^{2}-x+3x-1$.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Factor $6x^{2}-11x+3$.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(3x-1\right)\left(-2x+3\right)$ and $\left(2x-3\right)\left(3x-1\right)$ is $\left(2x-3\right)\left(3x-1\right)$. Multiply $\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}$ times $\frac{-1}{-1}$.

$$\frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)}+\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Since $\frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)}$ and $\frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-\left(x^{2}+x+5\right)+4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Do the multiplications in $-\left(x^{2}+x+5\right)+4x^{2}+6x+3$.

$$\frac{-x^{2}-x-5+4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)}$$

Combine like terms in $-x^{2}-x-5+4x^{2}+6x+3$.

$$\frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}$$

Factor the expressions that are not already factored in $\frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}$.

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

Cancel out $3x-1$ in both numerator and denominator.

$$\frac{x+2}{2x-3}$$