Math  /  Algebra

Question14. 928+3x+2=34\frac{9}{28}+\frac{3}{x+2}=\frac{3}{4}

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes this equation true! Watch out! Remember, we can't divide by zero, so keep an eye on that x+2x + 2!

STEP 2

1. Isolate the term with xx
2. Multiply to get rid of fractions
3. Solve for xx

STEP 3

Alright, let's **isolate** that term with xx in it!
We'll subtract 928\frac{9}{28} from both sides of the equation: 3x+2=34928 \frac{3}{x+2} = \frac{3}{4} - \frac{9}{28} Why? Because it **moves** the 928\frac{9}{28} to the other side and gets us closer to solving for xx!

STEP 4

Now, let's **simplify** the right side.
We need a common denominator, which is **28**: 3x+2=3747928=2128928=1228 \frac{3}{x+2} = \frac{3 \cdot 7}{4 \cdot 7} - \frac{9}{28} = \frac{21}{28} - \frac{9}{28} = \frac{12}{28} We multiplied the numerator and denominator of 34\frac{3}{4} by **7** to make the denominator **28**.
This allows us to subtract the fractions!
Simplifying the fraction gives us: 3x+2=1228=3474=37 \frac{3}{x+2} = \frac{12}{28} = \frac{3 \cdot 4}{7 \cdot 4} = \frac{3}{7} We divided both the numerator and the denominator by their **greatest common divisor**, which is **4**.

STEP 5

Fractions can be tricky, so let's get rid of them!
We can **multiply** both sides of the equation by 7(x+2)7(x+2).
This is allowed as long as x+2x+2 isn't zero, which we'll check later! 3x+27(x+2)=377(x+2) \frac{3}{x+2} \cdot 7(x+2) = \frac{3}{7} \cdot 7(x+2) 37=3(x+2) 3 \cdot 7 = 3(x+2) We can **divide to one** by canceling out the common factors of x+2x+2 on the left side and 77 on the right side.

STEP 6

Now, let's **simplify**: 21=3(x+2) 21 = 3(x+2)

STEP 7

To **isolate** x+2x+2, we'll **divide** both sides by **3**: 213=3(x+2)3 \frac{21}{3} = \frac{3(x+2)}{3} 7=x+2 7 = x+2

STEP 8

Almost there!
Let's **subtract 2** from both sides to finally **solve for** xx: 72=x+22 7 - 2 = x + 2 - 2 5=x 5 = x So, x=5x = \mathbf{5}!

STEP 9

x=5x = \mathbf{5}

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