Rational Expressions Question Stack Activity | Math = Love - Free Printable

Problem: Simplify the given rational expressions.

We will solve each expression step by step. Let's go through them one by one.

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#### 1. Simplify:
$$
\frac{3x - 1}{x + 2} - \frac{x - 2}{x - 1}
$$

Step 1: Find a common denominator.
The denominators are \(x + 2\) and \(x - 1\). The least common denominator (LCD) is:
$$
(x + 2)(x - 1)
$$

Step 2: Rewrite each fraction with the common denominator.
$$
\frac{3x - 1}{x + 2} = \frac{(3x - 1)(x - 1)}{(x + 2)(x - 1)}
$$
$$
\frac{x - 2}{x - 1} = \frac{(x - 2)(x + 2)}{(x - 1)(x + 2)}
$$

So, the expression becomes:
$$
\frac{(3x - 1)(x - 1)}{(x + 2)(x - 1)} - \frac{(x - 2)(x + 2)}{(x - 1)(x + 2)}
$$

Step 3: Combine the fractions.
$$
\frac{(3x - 1)(x - 1) - (x - 2)(x + 2)}{(x + 2)(x - 1)}
$$

Step 4: Expand the numerators.
For \((3x - 1)(x - 1)\):
$$
(3x - 1)(x - 1) = 3x^2 - 3x - x + 1 = 3x^2 - 4x + 1
$$

For \((x - 2)(x + 2)\):
$$
(x - 2)(x + 2) = x^2 - 4
$$

So the expression becomes:
$$
\frac{3x^2 - 4x + 1 - (x^2 - 4)}{(x + 2)(x - 1)}
$$

Step 5: Simplify the numerator.
$$
3x^2 - 4x + 1 - x^2 + 4 = 2x^2 - 4x + 5
$$

Thus, the simplified expression is:
$$
\frac{2x^2 - 4x + 5}{(x + 2)(x - 1)}
$$

Final Answer:
$$
\boxed{\frac{2x^2 - 4x + 5}{(x + 2)(x - 1)}}
$$

---

#### 2. Simplify:
$$
\frac{2}{x + 2} - \frac{1}{x - 1}
$$

Step 1: Find a common denominator.
The denominators are \(x + 2\) and \(x - 1\). The LCD is:
$$
(x + 2)(x - 1)
$$

Step 2: Rewrite each fraction with the common denominator.
$$
\frac{2}{x + 2} = \frac{2(x - 1)}{(x + 2)(x - 1)}
$$
$$
\frac{1}{x - 1} = \frac{1(x + 2)}{(x - 1)(x + 2)}
$$

So, the expression becomes:
$$
\frac{2(x - 1)}{(x + 2)(x - 1)} - \frac{1(x + 2)}{(x - 1)(x + 2)}
$$

Step 3: Combine the fractions.
$$
\frac{2(x - 1) - 1(x + 2)}{(x + 2)(x - 1)}
$$

Step 4: Expand the numerator.
$$
2(x - 1) = 2x - 2
$$
$$
1(x + 2) = x + 2
$$

So the expression becomes:
$$
\frac{2x - 2 - (x + 2)}{(x + 2)(x - 1)}
$$

Step 5: Simplify the numerator.
$$
2x - 2 - x - 2 = x - 4
$$

Thus, the simplified expression is:
$$
\frac{x - 4}{(x + 2)(x - 1)}
$$

Final Answer:
$$
\boxed{\frac{x - 4}{(x + 2)(x - 1)}}
$$

---

#### 3. Simplify:
$$
\frac{x + 1}{x - 3} + \frac{x}{x + 2}
$$

Step 1: Find a common denominator.
The denominators are \(x - 3\) and \(x + 2\). The LCD is:
$$
(x - 3)(x + 2)
$$

Step 2: Rewrite each fraction with the common denominator.
$$
\frac{x + 1}{x - 3} = \frac{(x + 1)(x + 2)}{(x - 3)(x + 2)}
$$
$$
\frac{x}{x + 2} = \frac{x(x - 3)}{(x + 2)(x - 3)}
$$

So, the expression becomes:
$$
\frac{(x + 1)(x + 2)}{(x - 3)(x + 2)} + \frac{x(x - 3)}{(x + 2)(x - 3)}
$$

Step 3: Combine the fractions.
$$
\frac{(x + 1)(x + 2) + x(x - 3)}{(x - 3)(x + 2)}
$$

Step 4: Expand the numerator.
For \((x + 1)(x + 2)\):
$$
(x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2
$$

For \(x(x - 3)\):
$$
x(x - 3) = x^2 - 3x
$$

So the expression becomes:
$$
\frac{x^2 + 3x + 2 + x^2 - 3x}{(x - 3)(x + 2)}
$$

Step 5: Simplify the numerator.
$$
x^2 + 3x + 2 + x^2 - 3x = 2x^2 + 2
$$

Thus, the simplified expression is:
$$
\frac{2x^2 + 2}{(x - 3)(x + 2)}
$$

Step 6: Factor the numerator (if possible).
$$
2x^2 + 2 = 2(x^2 + 1)
$$

So the expression becomes:
$$
\frac{2(x^2 + 1)}{(x - 3)(x + 2)}
$$

Final Answer:
$$
\boxed{\frac{2(x^2 + 1)}{(x - 3)(x + 2)}}
$$

---

#### 4. Simplify:
$$
\frac{x^2 - 4x + 4}{x^2 - 3x + 2}
$$

Step 1: Factor the numerator and the denominator.

- Numerator: \(x^2 - 4x + 4\)
This is a perfect square trinomial:
$$
x^2 - 4x + 4 = (x - 2)^2
$$

- Denominator: \(x^2 - 3x + 2\)
Factor by finding two numbers that multiply to \(2\) and add to \(-3\). These numbers are \(-1\) and \(-2\):
$$
x^2 - 3x + 2 = (x - 1)(x - 2)
$$

So the expression becomes:
$$
\frac{(x - 2)^2}{(x - 1)(x - 2)}
$$

Step 2: Cancel the common factor \((x - 2)\).
$$
\frac{(x - 2)^2}{(x - 1)(x - 2)} = \frac{x - 2}{x - 1} \quad \text{(for } x \neq 2\text{)}
$$

Final Answer:
$$
\boxed{\frac{x - 2}{x - 1}}
$$

---

#### 5. Simplify:
$$
\frac{x^2 - 1}{x^2 + x - 2}
$$

Step 1: Factor the numerator and the denominator.

- Numerator: \(x^2 - 1\)
This is a difference of squares:
$$
x^2 - 1 = (x - 1)(x + 1)
$$

- Denominator: \(x^2 + x - 2\)
Factor by finding two numbers that multiply to \(-2\) and add to \(1\). These numbers are \(2\) and \(-1\):
$$
x^2 + x - 2 = (x + 2)(x - 1)
$$

So the expression becomes:
$$
\frac{(x - 1)(x + 1)}{(x + 2)(x - 1)}
$$

Step 2: Cancel the common factor \((x - 1)\).
$$
\frac{(x - 1)(x + 1)}{(x + 2)(x - 1)} = \frac{x + 1}{x + 2} \quad \text{(for } x \neq 1\text{)}
$$

Final Answer:
$$
\boxed{\frac{x + 1}{x + 2}}
$$

---

#### 6. Simplify:
$$
\frac{x + 3}{x + 5} + \frac{6}{x^2 + 3x - 10}
$$

Step 1: Factor the denominator of the second fraction.

The denominator \(x^2 + 3x - 10\) can be factored as:
$$
x^2 + 3x - 10 = (x + 5)(x - 2)
$$

So the expression becomes:
$$
\frac{x + 3}{x + 5} + \frac{6}{(x + 5)(x - 2)}
$$

Step 2: Find a common denominator.
The denominators are \(x + 5\) and \((x + 5)(x - 2)\). The LCD is:
$$
(x + 5)(x - 2)
$$

Step 3: Rewrite each fraction with the common denominator.
$$
\frac{x + 3}{x + 5} = \frac{(x + 3)(x - 2)}{(x + 5)(x - 2)}
$$
$$
\frac{6}{(x + 5)(x - 2)} = \frac{6}{(x + 5)(x - 2)}
$$

So the expression becomes:
$$
\frac{(x + 3)(x - 2)}{(x + 5)(x - 2)} + \frac{6}{(x + 5)(x - 2)}
$$

Step 4: Combine the fractions.
$$
\frac{(x + 3)(x - 2) + 6}{(x + 5)(x - 2)}
$$

Step 5: Expand the numerator.
For \((x + 3)(x - 2)\):
$$
(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
$$

So the expression becomes:
$$
\frac{x^2 + x - 6 + 6}{(x + 5)(x - 2)}
$$

Step 6: Simplify the numerator.
$$
x^2 + x - 6 + 6 = x^2 + x
$$

Thus, the simplified expression is:
$$
\frac{x^2 + x}{(x + 5)(x - 2)}
$$

Step 7: Factor the numerator (if possible).
$$
x^2 + x = x(x + 1)
$$

So the expression becomes:
$$
\frac{x(x + 1)}{(x + 5)(x - 2)}
$$

Final Answer:
$$
\boxed{\frac{x(x + 1)}{(x + 5)(x - 2)}}
$$

---

#### 7. Simplify:
$$
\frac{4x^2 - 16}{2 - x}
$$

Step 1: Factor the numerator.

The numerator \(4x^2 - 16\) is a difference of squares:
$$
4x^2 - 16 = 4(x^2 - 4) = 4(x - 2)(x + 2)
$$

So the expression becomes:
$$
\frac{4(x - 2)(x + 2)}{2 - x}
$$

Step 2: Rewrite \(2 - x\) as \(-(x - 2)\).
$$
2 - x = -(x - 2)
$$

So the expression becomes:
$$
\frac{4(x - 2)(x + 2)}{-(x - 2)}
$$

Step 3: Cancel the common factor \((x - 2)\).
$$
\frac{4(x - 2)(x + 2)}{-(x - 2)} = \frac{4(x + 2)}{-1} = -4(x + 2)
$$

Final Answer:
$$
\boxed{-4(x + 2)}
$$

---

#### 8. Simplify:
$$
\frac{24x^2 - 40x^3}{8x}
$$

Step 1: Factor the numerator.

Factor out the greatest common factor (GCF) from the numerator:
$$
24x^2 - 40x^3 = 8x^2(3 - 5x)
$$

So the expression becomes:
$$
\frac{8x^2(3 - 5x)}{8x}
$$

Step 2: Cancel the common factors.
$$
\frac{8x^2(3 - 5x)}{8x} = \frac{8x \cdot x(3 - 5x)}{8x} = x(3 - 5x)
$$

Final Answer:
$$
\boxed{x(3 - 5x)}
$$

---

#### 9. Simplify:
$$
\frac{4x^2 + 4x - 8}{x + 2}
$$

Step 1: Factor the numerator.

Factor out the GCF from the numerator:
$$
4x^2 + 4x - 8 = 4(x^2 + x - 2)
$$

Now factor \(x^2 + x - 2\):
$$
x^2 + x - 2 = (x + 2)(x - 1)
$$

So the numerator becomes:
$$
4(x^2 + x - 2) = 4(x + 2)(x - 1)
$$

The expression now is:
$$
\frac{4(x + 2)(x - 1)}{x + 2}
$$

Step 2: Cancel the common factor \((x + 2)\).
$$
\frac{4(x + 2)(x - 1)}{x + 2} = 4(x - 1)
$$

Final Answer:
$$
\boxed{4(x - 1)}
$$

---

#### 10. Simplify:
$$
\frac{x^2 + x - 6}{x^2 + 5x + 6}
$$

Step 1: Factor the numerator and the denominator.

- Numerator: \(x^2 + x - 6\)
Factor by finding two numbers that multiply to \(-6\) and add to \(1\). These numbers are \(3\) and \(-2\):
$$
x^2 + x - 6 = (x + 3)(x - 2)
$$

- Denominator: \(x^2 + 5x + 6\)
Factor by finding two numbers that multiply to \(6\) and add to \(5\). These numbers are \(3\) and \(2\):
$$
x^2 + 5x + 6 = (x + 3)(x + 2)
$$

So the expression becomes:
$$
\frac{(x + 3)(x - 2)}{(x + 3)(x + 2)}
$$

Step 2: Cancel the common factor \((x + 3)\).
$$
\frac{(x + 3)(x - 2)}{(x + 3)(x + 2)} = \frac{x - 2}{x + 2} \quad \text{(for } x \neq -3\text{)}
$$

Final Answer:
$$
\boxed{\frac{x - 2}{x + 2}}
$$

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Final Answers:

1. \(\boxed{\frac{2x^2 - 4x + 5}{(x + 2)(x - 1)}}\)
2. \(\boxed{\frac{x - 4}{(x + 2)(x - 1)}}\)
3. \(\boxed{\frac{2(x^2 + 1)}{(x - 3)(x + 2)}}\)
4. \(\boxed{\frac{x - 2}{x - 1}}\)
5. \(\boxed{\frac{x + 1}{x + 2}}\)
6. \(\boxed{\frac{x(x + 1)}{(x + 5)(x - 2)}}\)
7. \(\boxed{-4(x + 2)}\)
8. \(\boxed{x(3 - 5x)}\)
9. \(\boxed{4(x - 1)}\)
10. \(\boxed{\frac{x - 2}{x + 2}}\)