How to subtract complex fractions - SSAT Upper Level Quantitative

Card 0 of 28

Question

Simplify.

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

Answer

Find the least common denominator which is .

Just multiply the right fraction top and bottom by .

$\frac{2}{x^2}-\frac{4\cdot x}{x \cdot x}=\frac{2}{x^2}-\frac{4x}{x^2}$

Finally, subtract.

Answer should be,

$\frac{2-4x}{x^2}$ .

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Question

Simplify.

$\frac{x^2-5x+6}{x^2+2x-8}-\frac{x^2-5x-6}{x^2+x-12}$

Answer

Lets try to factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

$\frac{(x-2)(x-3)}{(x-2)(x+4)}-\frac{(x-3)(x-2)}{(x-3)(x+4)}$

Next cancel out the like terms.

$\frac{x-3}{x+4}-\frac{x-2}{x+4}$

Now combine the numerator. Remember to distribute the negative sign.

This is the final answer:

$\frac{-1}{x+4}$ .

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Question

Simplify.

$1-\frac{3}{x-3}$

Answer

Turn the into a fraction that has a common denominator with the other fraction. To do this multiply $1 \cdot \frac{x-3}{x-3}$ .

This results in the following expression:

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

With the same denominator, just subtract and remember to distribute the negative sign.

The final answer is

$\frac{x-6}{x-3}$ .

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Question

Solve and simplify.

$\frac{x^2-9}{x+3}-\frac{x^2-16}{x+4}$

Answer

Lets try to reduce the fraction. When factoring a difference of squares, the form is . When you foil, the middle terms cancel out.

So when doing that, we have:

$\frac{(x-3)(x+3)}{(x+3)}-\frac{(x-4)(x+4)}{(x+4)}$ .

Cancel out like terms and we get this:

.

Distribute the negative sign and we should get .

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Question

Solve for .

$\frac{1}{\sqrt{x}}-\frac{1}{2\sqrt{x}}=2$

Answer

First find the least common denominator. That will be $2\sqrt{x}$ .

Multiply top and bottom of the left fraction by .

$\frac{1}{\sqrt{x}}\cdot \frac{2}{2}=\frac{2}{2\sqrt{x}}$

Then subtract the numerator and then cross-multiply to get this:

$\frac{2}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}=2\rightarrow 2-1=2(2\sqrt{x}) \rightarrow 1=4\sqrt{x}$

Then combine like terms. To get rid of the square root,

$\frac{1}{4}=\sqrt{x}$ square both sides to get $\left(\frac{1}{4}\right)^2=\frac{1}{16}$ as the final answer.

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Question

Solve for .

$\frac{5}{x-2}-\frac{2}{x+2}=1$

Answer

Find the least common denominator which is . Then multiply the left fraction numerator by and multiply the right numerator by inorder for each fraction to share the common denominator.

$\frac{5(x+2)}{x^2-4}-\frac{2(x-2)}{x^2-4}=1$

Distribute and be careful of the negative sign.

$\frac{5x+10-2x+4}{x^2-4}=1$

Cross-multiply and create the quadratic equation.

Lets factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

,

If you check these answers back into the question, none of the fractions are undefined so these are the final answers.

Compare your answer with the correct one above

Question

Solve for .

$\frac{2}{x-1}-\frac{2}{x+1}=\left | 2\right |$

Answer

Find the least common denominator which is . Then multiply the left fraction numerator by and multiply the right numerator by in order to make each fraction share the common denominator.

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

Distribute and be careful of the negative sign.

$\frac{2x+2-2x+2}{x^2-1}=\left | 2\right |$

Simplify the numerator and cross multiply. Because there is an absolute value bar, we need to split this expression into two different equations.

Equation one:

$x=\pm \sqrt{3}$ By inspection, these values pass and don't violate the fractions being undefined.

Equation two:

This answer is imaginary and not in the choices leaving $\pm \sqrt{3}$ as the answers.

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Question

Simplify.

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

Answer

Find the least common denominator which is .

Just multiply the right fraction top and bottom by .

$\frac{2}{x^2}-\frac{4\cdot x}{x \cdot x}=\frac{2}{x^2}-\frac{4x}{x^2}$

Finally, subtract.

Answer should be,

$\frac{2-4x}{x^2}$ .

Compare your answer with the correct one above

Question

Simplify.

$\frac{x^2-5x+6}{x^2+2x-8}-\frac{x^2-5x-6}{x^2+x-12}$

Answer

Lets try to factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

$\frac{(x-2)(x-3)}{(x-2)(x+4)}-\frac{(x-3)(x-2)}{(x-3)(x+4)}$

Next cancel out the like terms.

$\frac{x-3}{x+4}-\frac{x-2}{x+4}$

Now combine the numerator. Remember to distribute the negative sign.

This is the final answer:

$\frac{-1}{x+4}$ .

Compare your answer with the correct one above

Question

Simplify.

$1-\frac{3}{x-3}$

Answer

Turn the into a fraction that has a common denominator with the other fraction. To do this multiply $1 \cdot \frac{x-3}{x-3}$ .

This results in the following expression:

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

With the same denominator, just subtract and remember to distribute the negative sign.

The final answer is

$\frac{x-6}{x-3}$ .

Compare your answer with the correct one above

Question

Solve and simplify.

$\frac{x^2-9}{x+3}-\frac{x^2-16}{x+4}$

Answer

Lets try to reduce the fraction. When factoring a difference of squares, the form is . When you foil, the middle terms cancel out.

So when doing that, we have:

$\frac{(x-3)(x+3)}{(x+3)}-\frac{(x-4)(x+4)}{(x+4)}$ .

Cancel out like terms and we get this:

.

Distribute the negative sign and we should get .

Compare your answer with the correct one above

Question

Solve for .

$\frac{1}{\sqrt{x}}-\frac{1}{2\sqrt{x}}=2$

Answer

First find the least common denominator. That will be $2\sqrt{x}$ .

Multiply top and bottom of the left fraction by .

$\frac{1}{\sqrt{x}}\cdot \frac{2}{2}=\frac{2}{2\sqrt{x}}$

Then subtract the numerator and then cross-multiply to get this:

$\frac{2}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}=2\rightarrow 2-1=2(2\sqrt{x}) \rightarrow 1=4\sqrt{x}$

Then combine like terms. To get rid of the square root,

$\frac{1}{4}=\sqrt{x}$ square both sides to get $\left(\frac{1}{4}\right)^2=\frac{1}{16}$ as the final answer.

Compare your answer with the correct one above

Question

Solve for .

$\frac{5}{x-2}-\frac{2}{x+2}=1$

Answer

Find the least common denominator which is . Then multiply the left fraction numerator by and multiply the right numerator by inorder for each fraction to share the common denominator.

$\frac{5(x+2)}{x^2-4}-\frac{2(x-2)}{x^2-4}=1$

Distribute and be careful of the negative sign.

$\frac{5x+10-2x+4}{x^2-4}=1$

Cross-multiply and create the quadratic equation.

Lets factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

,

If you check these answers back into the question, none of the fractions are undefined so these are the final answers.

Compare your answer with the correct one above

Question

Solve for .

$\frac{2}{x-1}-\frac{2}{x+1}=\left | 2\right |$

Answer

Find the least common denominator which is . Then multiply the left fraction numerator by and multiply the right numerator by in order to make each fraction share the common denominator.

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

Distribute and be careful of the negative sign.

$\frac{2x+2-2x+2}{x^2-1}=\left | 2\right |$

Simplify the numerator and cross multiply. Because there is an absolute value bar, we need to split this expression into two different equations.

Equation one:

$x=\pm \sqrt{3}$ By inspection, these values pass and don't violate the fractions being undefined.

Equation two:

This answer is imaginary and not in the choices leaving $\pm \sqrt{3}$ as the answers.

Compare your answer with the correct one above

Question

Simplify.

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

Answer

Find the least common denominator which is .

Just multiply the right fraction top and bottom by .

$\frac{2}{x^2}-\frac{4\cdot x}{x \cdot x}=\frac{2}{x^2}-\frac{4x}{x^2}$

Finally, subtract.

Answer should be,

$\frac{2-4x}{x^2}$ .

Compare your answer with the correct one above

Question

Simplify.

$\frac{x^2-5x+6}{x^2+2x-8}-\frac{x^2-5x-6}{x^2+x-12}$

Answer

Lets try to factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

$\frac{(x-2)(x-3)}{(x-2)(x+4)}-\frac{(x-3)(x-2)}{(x-3)(x+4)}$

Next cancel out the like terms.

$\frac{x-3}{x+4}-\frac{x-2}{x+4}$

Now combine the numerator. Remember to distribute the negative sign.

This is the final answer:

$\frac{-1}{x+4}$ .

Compare your answer with the correct one above

Question

Simplify.

$1-\frac{3}{x-3}$

Answer

Turn the into a fraction that has a common denominator with the other fraction. To do this multiply $1 \cdot \frac{x-3}{x-3}$ .

This results in the following expression:

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

With the same denominator, just subtract and remember to distribute the negative sign.

The final answer is

$\frac{x-6}{x-3}$ .

Compare your answer with the correct one above

Question

Solve and simplify.

$\frac{x^2-9}{x+3}-\frac{x^2-16}{x+4}$

Answer

Lets try to reduce the fraction. When factoring a difference of squares, the form is . When you foil, the middle terms cancel out.

So when doing that, we have:

$\frac{(x-3)(x+3)}{(x+3)}-\frac{(x-4)(x+4)}{(x+4)}$ .

Cancel out like terms and we get this:

.

Distribute the negative sign and we should get .

Compare your answer with the correct one above

Question

Solve for .

$\frac{1}{\sqrt{x}}-\frac{1}{2\sqrt{x}}=2$

Answer

First find the least common denominator. That will be $2\sqrt{x}$ .

Multiply top and bottom of the left fraction by .

$\frac{1}{\sqrt{x}}\cdot \frac{2}{2}=\frac{2}{2\sqrt{x}}$

Then subtract the numerator and then cross-multiply to get this:

$\frac{2}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}=2\rightarrow 2-1=2(2\sqrt{x}) \rightarrow 1=4\sqrt{x}$

Then combine like terms. To get rid of the square root,

$\frac{1}{4}=\sqrt{x}$ square both sides to get $\left(\frac{1}{4}\right)^2=\frac{1}{16}$ as the final answer.

Compare your answer with the correct one above

Question

Solve for .

$\frac{5}{x-2}-\frac{2}{x+2}=1$

Answer

Find the least common denominator which is . Then multiply the left fraction numerator by and multiply the right numerator by inorder for each fraction to share the common denominator.

$\frac{5(x+2)}{x^2-4}-\frac{2(x-2)}{x^2-4}=1$

Distribute and be careful of the negative sign.

$\frac{5x+10-2x+4}{x^2-4}=1$

Cross-multiply and create the quadratic equation.

Lets factor. Remember, we need to find two terms that are factors of the c term that add up to the b term.

,

If you check these answers back into the question, none of the fractions are undefined so these are the final answers.

Compare your answer with the correct one above

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