Question

1. solve and check for extraneous solutions:

show complete work on your paper!
\\(\frac{x - 5}{4}+\frac{x - 4}{5}=3\\)
solution: \\(x = \square\\) (leave answer as a reduced fraction... no decimal approximations allowed.)
\\(\bigcirc\\) this solution is not extraneous.
\\(\bigcirc\\) this solution is extraneous.

Explanation:

Step1: Eliminate denominators

Multiply all terms by $20$ (the least common multiple of 4 and 5):
$$5(x-5) + 4(x-4) = 3 \times 20$$

Step2: Expand parentheses

Distribute the coefficients:
$$5x - 25 + 4x - 16 = 60$$

Step3: Combine like terms

Combine variable and constant terms:
$$9x - 41 = 60$$

Step4: Isolate the variable term

Add 41 to both sides:
$$9x = 60 + 41$$
$$9x = 101$$

Step5: Solve for $x$

Divide both sides by 9:
$$x = \frac{101}{9}$$

Step6: Check for extraneous solutions

Verify $x=\frac{101}{9}$ does not make original denominators zero (denominators are 4 and 5, which are never zero), so it is valid.

Answer:

$x = \frac{101}{9}$
○ This solution is not extraneous.