Question

i can find the solutions to rational equations and ignore extraneous solutions.

1. find all values of x that make the equation true. \\(\frac{3}{x - 4} = \frac{x - 5}{x}\\) 3 points

Explanation:

Step1: Cross - multiply the equation

Given the rational equation \(\frac{3}{x - 4}=\frac{x - 5}{x}\), cross - multiplying (which is valid when \(x
eq0\) and \(x
eq4\) to avoid division by zero) gives us \(3\times x=(x - 5)\times(x - 4)\).
So, \(3x=(x - 5)(x - 4)\).

Step2: Expand the right - hand side

Expand \((x - 5)(x - 4)\) using the FOIL method. \((x-5)(x - 4)=x\times x-x\times4-5\times x + 5\times4=x^{2}-4x-5x + 20=x^{2}-9x + 20\).
Now our equation is \(3x=x^{2}-9x + 20\).

Step3: Rearrange the equation to form a quadratic equation

Subtract \(3x\) from both sides of the equation: \(x^{2}-9x + 20-3x = 0\).
Simplify the left - hand side: \(x^{2}-12x + 20 = 0\).

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \(20\) and add up to \(- 12\). The numbers are \(-10\) and \(-2\) since \((-10)\times(-2)=20\) and \(-10+( - 2)=-12\).
So, \(x^{2}-12x + 20=(x - 10)(x - 2)=0\).

Step5: Solve for x

Set each factor equal to zero:

• If \(x - 10=0\), then \(x = 10\).
• If \(x - 2=0\), then \(x = 2\).

Step6: Check for extraneous solutions

We need to check if these solutions make the original denominators zero.

• For \(x = 10\): The denominator \(x-4=10 - 4 = 6

eq0\) and \(x = 10
eq0\).

• For \(x = 2\): The denominator \(x - 4=2-4=-2

eq0\) and \(x = 2
eq0\).

Answer:

The values of \(x\) that make the equation true are \(x = 2\) and \(x = 10\).