Solve 2m-9/2m-5-3m/2-3m-1

$$\frac { 2 m - 9 } { 2 m - 5 } - \frac { 3 m } { 2 - 3 m } - 1$$

Evaluate

$$\frac{-6m^{2}+27m-8}{\left(2-3m\right)\left(2m-5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2m-5 and 2-3m is \left(2m-5\right)\left(-3m+2\right). Multiply \frac{2m-9}{2m-5} times \frac{-3m+2}{-3m+2}. Multiply \frac{3m}{2-3m} times \frac{2m-5}{2m-5}.

$$\frac{\left(2m-9\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}-\frac{3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Since \frac{\left(2m-9\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)} and \frac{3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2m-9\right)\left(-3m+2\right)-3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Do the multiplications in \left(2m-9\right)\left(-3m+2\right)-3m\left(2m-5\right).

$$\frac{-6m^{2}+4m+27m-18-6m^{2}+15m}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Combine like terms in -6m^{2}+4m+27m-18-6m^{2}+15m.

$$\frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}.

$$\frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)}-\frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}$$

Since \frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)} and \frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-12m^{2}+46m-18-\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}$$

Do the multiplications in -12m^{2}+46m-18-\left(2m-5\right)\left(-3m+2\right).

$$\frac{-12m^{2}+46m-18+6m^{2}-4m-15m+10}{\left(2m-5\right)\left(-3m+2\right)}$$

Combine like terms in -12m^{2}+46m-18+6m^{2}-4m-15m+10.

$$\frac{-6m^{2}+27m-8}{\left(2m-5\right)\left(-3m+2\right)}$$

Expand \left(2m-5\right)\left(-3m+2\right).

$$\frac{-6m^{2}+27m-8}{-6m^{2}+19m-10}$$

Expand

$$\frac{-6m^{2}+27m-8}{\left(2-3m\right)\left(2m-5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2m-5 and 2-3m is \left(2m-5\right)\left(-3m+2\right). Multiply \frac{2m-9}{2m-5} times \frac{-3m+2}{-3m+2}. Multiply \frac{3m}{2-3m} times \frac{2m-5}{2m-5}.

$$\frac{\left(2m-9\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}-\frac{3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Since \frac{\left(2m-9\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)} and \frac{3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2m-9\right)\left(-3m+2\right)-3m\left(2m-5\right)}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Do the multiplications in \left(2m-9\right)\left(-3m+2\right)-3m\left(2m-5\right).

$$\frac{-6m^{2}+4m+27m-18-6m^{2}+15m}{\left(2m-5\right)\left(-3m+2\right)}-1$$

Combine like terms in -6m^{2}+4m+27m-18-6m^{2}+15m.

$$\frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}.

$$\frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)}-\frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}$$

Since \frac{-12m^{2}+46m-18}{\left(2m-5\right)\left(-3m+2\right)} and \frac{\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-12m^{2}+46m-18-\left(2m-5\right)\left(-3m+2\right)}{\left(2m-5\right)\left(-3m+2\right)}$$

Do the multiplications in -12m^{2}+46m-18-\left(2m-5\right)\left(-3m+2\right).

$$\frac{-12m^{2}+46m-18+6m^{2}-4m-15m+10}{\left(2m-5\right)\left(-3m+2\right)}$$

Combine like terms in -12m^{2}+46m-18+6m^{2}-4m-15m+10.

$$\frac{-6m^{2}+27m-8}{\left(2m-5\right)\left(-3m+2\right)}$$

Expand \left(2m-5\right)\left(-3m+2\right).

$$\frac{-6m^{2}+27m-8}{-6m^{2}+19m-10}$$