A. Solve the following rational equations.Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (20 points)

2.  x        5         1
    ----   -   ----  = - -----
     4       2x       4x 

Solution Answer:

Checking:

Let's solve the given rational equation step by step. The equation is:

\[
\frac{x}{4} - \frac{5}{2x} = \frac{1}{4x}
\]

### Step 1: Identify the Least Common Denominator (LCD)
The denominators are \( 4 \), \( 2x \), and \( 4x \). The least common denominator (LCD) of these three terms is \( 4x \), as it covers all factors present in the denominators.

### Step 2: Multiply every term by the LCD
To eliminate the denominators, we multiply every term in the equation by \( 4x \):

\[
4x \cdot \frac{x}{4} - 4x \cdot \frac{5}{2x} = 4x \cdot \frac{1}{4x}
\]

This simplifies as follows:

\[
x \cdot x - 2 \cdot 5 = 1
\]

\[
x^2 - 10 = 1
\]

### Step 3: Solve for \( x \)
Now, solve the quadratic equation:

\[
x^2 - 10 = 1
\]

Add 10 to both sides:

\[
x^2 = 11
\]

Take the square root of both sides:

\[
x = \pm \sqrt{11}
\]

So the solutions are:

\[
x = \sqrt{11} \quad \text{or} \quad x = -\sqrt{11}
\]

### Step 4: Check the Solutions
Now we need to check if these values satisfy the original equation. 

#### Checking \( x = \sqrt{11} \):

Substitute \( x = \sqrt{11} \) into the original equation:

\[
\frac{\sqrt{11}}{4} - \frac{5}{2\sqrt{11}} = \frac{1}{4\sqrt{11}}
\]

Multiply both sides by \( 4\sqrt{11} \) to simplify:

\[
\sqrt{11} \cdot \sqrt{11} - 2 \cdot 5 = 1
\]

\[
11 - 10 = 1
\]

This is true, so \( x = \sqrt{11} \) is a valid solution.

#### Checking \( x = -\sqrt{11} \):

Substitute \( x = -\sqrt{11} \) into the original equation:

\[
\frac{-\sqrt{11}}{4} - \frac{5}{2(-\sqrt{11})} = \frac{1}{4(-\sqrt{11})}
\]

Again, multiply both sides by \( 4\sqrt{11} \):

\[
-\sqrt{11} \cdot \sqrt{11} - (-10) = 1
\]

\[
-11 + 10 = -1 \quad \text{which is false.}
\]

So, \( x = -\sqrt{11} \) is **not** a valid solution.

### Final Answer:
The only valid solution is:

\[
x = \sqrt{11}
\]

### Summary:

- LCD: \( 4x \)
- Correct steps: Yes
- Solutions: \( x = \sqrt{11} \)
- Checking: Valid solution is \( x = \sqrt{11} \), \( x = -\sqrt{11} \) is not valid.

### Do you have any questions or want further details?

Here are 5 related questions:
1. How do you find the least common denominator in rational equations?
2. How do you handle square roots when solving equations?
3. What are common pitfalls when checking solutions in rational equations?
4. Why did multiplying by the LCD simplify the equation?
5. What happens if a solution makes the denominator zero?

**Tip**: Always check for restrictions on \( x \) that might make a denominator zero when solving rational equations.

Solve the following rational equations. Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (20 points)

2.  x/4 - 5/2x = 1/4x

Master solving rational equations with this detailed example: x/4 - 5/2x = 1/4x. Learn how to find the least common denominator (LCD), apply quadratic equation principles, and verify solutions step by step. Suitable for students in Grades 10-12.

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