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Pre-Algebra : Two-Step Equations with Fractions

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Example Questions

Example Question #1 : Two Step Equations With Fractions

Solve for :

$\frac{1}{2}x+\frac{3}{4}=9$

Possible Answers:

$16\frac{3}{4}$

$16\frac{1}{4}$

$4\frac{1}{2}$

$16\frac{1}{2}$

Correct answer:

$16\frac{1}{2}$

Explanation:

The goal is to isolate the variable on one side.

$\frac{1}{2}x+\frac{3}{4}=9$

Subtract $\frac{3}{4 }$ from each side of the equation:

$\frac{1}{2}x+\frac{3}{4}-\frac{3}{4}=9-\frac{3}{4}$

$\frac{1}{2}x=8\frac{1}{4}$

Multiply both sides by :

$x=16\frac{1}{2}$

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Example Question #2 : Two Step Equations With Fractions

Solve for :

$\frac{2}{3}x+6\frac{1}{3}=10\frac{1}{3}$

Possible Answers:

Correct answer:

Explanation:

The goal is to isolate the variable to one side.

$\frac{2}{3}x+6\frac{1}{3}=10\frac{1}{3}$

First, convert mixed numbers to improper fractions:

$\frac{2}{3}x+\frac{19}{3}=\frac{31}{3}$

Subtract $\frac{19}{3}$ from both sides:

$\frac{2}{3}x+\frac{19}{3}-\frac{19}{3}=\frac{31}{3}-\frac{19}{3}$

$\frac{2}{3}x=\frac{12}{3}$

Multiply each side by the reciprocal of $\frac{2}{3}$ :

$\frac{3}{2}*\frac{2}{3}x=\frac{12}{3}*\frac{3}{2}$

Cross out like terms and multiply:

$\frac{1}{1}x=\frac{6}{1}*\frac{1}{1}$

x=6

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Example Question #2 : Two Step Equations With Fractions

Solve for :

$\frac{2}{3}x-\frac{1}{4}=\frac{1}{6}$

Possible Answers:

$\frac{5}{12}$

$\frac{5}{18}$

$-\frac{1}{18}$

$-\frac{1}{8}$

$\frac{5}{8}$

Correct answer:

$\frac{5}{8}$

Explanation:

Step 1: Add $\frac{1}{4}$ to both sides:

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

$\frac{2}{3}x+0=\frac{1}{6}+\frac{1}{4}$

$\frac{2}{3}x=\frac{1}{6}+\frac{1}{4}$

Step 2: Add $\frac{1}{6}$ to $\frac{1}{4}$ . Remember that when you add fractions, you must find common denominator. The common denominator for $\frac{1}{4}$ and $\frac{1}{6}$ is . $\frac{1}{6}$ becomes $\frac{2}{12}$ when you multiply both the numerator and the denominator by . Similarly, $\frac{1}{4}$ becomes $\frac{3}{12}$ when you multiply both the numerator and the denominator by .

$\frac{2}{3}x=\frac{1}{6}+\frac{1}{4}$

$\frac{2}{3}x=\frac{2}{12}+\frac{3}{12}$

$\frac{2}{3}x=\frac{5}{12}$

Step 3: Multiply both sides of the equation by the reciprocal of $\frac{2}{3}$ :

$\frac{3}{2}\times \frac{2}{3}x=\frac{5}{12}\times \frac{3}{2}$

$1x=\frac{15}{24}$

$x=\frac{15}{24}$

Step 4: Simplify the fraction by dividing the numerator and the denominator by the Greatest Common Factor (GCF). The GCF of and is :

$x=\frac{15}{24}=\frac{\frac{15}{3}}{\frac{24}{3}}=\frac{5}{8}$

$x=\frac{5}{8}$

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Example Question #3 : Two Step Equations With Fractions

Solve for $\small x$ :

$\frac{1}{2}x+2=4$

Possible Answers:

$\small 4$

$\small 8$

$\small 6$

$\small 1$

$\small 2$

Correct answer:

$\small 4$

Explanation:

You are trying to isolate the $\small x$ .

To do this you must first subtract both sides by 2 to get

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

This then becomes a one-step problem where you multiply both sides by 2 to get

x=4

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Example Question #3 : Two Step Equations With Fractions

Solve for x:

$x-\frac{1}{4}=\frac{1}{2}$

Possible Answers:

x=1

$x=\frac{3}{8}$

$x=\frac{3}{2}$

$x=-\frac{1}{4}$

$x=\frac{3}{4}$

Correct answer:

$x=\frac{3}{4}$

Explanation:

Once you've isolated x, it's important to find the lowest common denominator so that you can add the two fractions you're working with.

Step 1: Isolate x and convert fractions so that they have a common denominator

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

$x=\frac{1}{2}+ \frac{1}{4}=\frac{1\cdot 2}{2\cdot 2} +\frac{1}{4}=\frac{2}{4}+\frac{1}{4}$

Step 2: solve for x

$x=\frac{2}{4}+\frac{1}{4}=\frac{3}{4}, x=\frac{3}{4}$

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Example Question #3 : Two Step Equations With Fractions

Solve for .

$\frac{x}4=\frac{9}x{}{}$

Possible Answers:

$x=\frac{36}x{}$

x=36

$x=\pm \frac{3}2{}$

$x=\frac{4}9{}$

$x=\pm 6$

Correct answer:

$x=\pm 6$

Explanation:

Cross multiplication is a short-cut that comes from multiplying by the denominators on both sides of an equation. Broken down, it works like this:

$4\cdot \frac{x}4=\frac{9}x\cdot 4{}{}$

The 4's on the left side of the equation cancel out.

$x=\frac{9\cdot 4}x{}$

Now, do the same with the denominator on the right side.

$x\cdot x=\frac{9\cdot 4}x\cdot x{}$

The 's on the right cancel out.

$x\cdot x=9\cdot 4$

This is simply the result of removing the denominators, then multiplying them on the opposite sides, i.e. cross multiplication.

Now, to finish solving for , simplify both sides.

$x^{2}=36$

Then take the square root to finish.

$x=\sqrt{36}$

$x=\pm 6$

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Example Question #6 : Two Step Equations With Fractions

Solve for :

$\frac{2}{3}s+\frac{1}{3}=-\frac{1}{3}$

Possible Answers:

s=2

s=1

s=-1

s=0

Correct answer:

s=-1

Explanation:

$\frac{2}{3}s+\frac{1}{3}=-\frac{1}{3}$

$\frac{2}{3}s+\frac{1}{3}-\frac{1}{3}=-\frac{1}{3}-\frac{1}{3}$

$\frac{2}{3}s=-\frac{2}{3}$

$\frac{3}{2}\cdot \left(\frac{2}{3}s\right)=\frac{3}{2}\cdot \left(-\frac{2}{3}\right)$

s=-1

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Example Question #1 : Two Step Equations With Fractions

Solve for .

$\frac{2}{5}x + \frac{4}{3} = 2$

Possible Answers:

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

x=2

$x=\frac{5}{3}$

$x=\frac{2}{3}$

$x=\frac{5}{6}$

Correct answer:

$x=\frac{5}{3}$

Explanation:

First, you want to leave all terms with x on one side and all other terms on the other side. To do this, we can subtract 4/3 from both sides.

We now have

$\frac{2}{5}x = \frac{2}{3}$

We can now multiply both sides by the reciprocal of 2/5, which is 5/2, to be able to solve for just x.

$x = \frac{2}{3} * \frac{5}{2} = \frac{5}{3}$

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Example Question #4 : Two Step Equations With Fractions

Solve for :

$1\frac{1}{3}x+7\frac{1}{3}=15\frac{1}{3}$

Possible Answers:

$6\frac{1}{3}$

$6\frac{2}{3}$

Correct answer:

Explanation:

Explanation:

The goal is to isolate the variable to one side.

$1\frac{1}{3}x+7\frac{1}{3}=15\frac{1}{3}$

First, convert the mixed numbers to improper fractions:

$\frac{4}{3}x+\frac{22}{3}=\frac{46}{3}$

Subtract $\frac{22}{3}$ from both sides:

$\frac{4}{3}x+\frac{22}{3}-\frac{22}{3}=\frac{46}{3}-\frac{22}{3}$

$\frac{4}{3}x=\frac{24}{3}$

Multiply each side by the reciprocal of $\frac{3}{4}$ :

$\frac{3}{4}*\frac{4}{3}x=\frac{24}{3}*\frac{3}{4}$

x=6

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Example Question #5 : Two Step Equations With Fractions

Solve for ""

$\frac{1}{2}x -8=4x-1$

Possible Answers:

x=5

x=4

x=-2

x=2

x=-3

Correct answer:

x=-2

Explanation:

1.) Add 8 to both sides, removing the "". It now reads $\frac{1}{2}x=4x+7$

2.) Multiply both sides by 2, removing the $\frac{1}{2}$ . It now reads x=8x+14

3.) Subtract from both sides, removing the "". It now reads -7x=14

4.) Divide both sides by "", resulting in x=-2

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Pre-Algebra : Two-Step Equations with Fractions

Study concepts, example questions & explanations for Pre-Algebra

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