Question

1. solve, state non - permissible values: $\frac{2}{5}-\frac{4}{5m}=\frac{30}{5m^{2}}$

Explanation:

Step1: Identify non-permissible values

Non-permissible values are values that make any denominator equal to 0.
Denominators are $5$, $5m$, and $5m^2$.
$5
eq 0$ for all $m$; $5m=0$ when $m=0$; $5m^2=0$ when $m=0$.
So non-permissible value: $m
eq 0$

Step2: Eliminate denominators

Multiply all terms by $5m^2$ (the lowest common denominator) to clear fractions:
$$5m^2 \cdot \frac{2}{5} - 5m^2 \cdot \frac{4}{5m} = 5m^2 \cdot \frac{30}{5m^2}$$
Simplify each term:
$$2m^2 - 4m = 30$$

Step3: Simplify the quadratic equation

Subtract 30 from both sides to set equation to 0:
$$2m^2 - 4m - 30 = 0$$
Divide all terms by 2 to simplify:
$$m^2 - 2m - 15 = 0$$

Step4: Factor the quadratic

Find two numbers that multiply to $-15$ and add to $-2$: $-5$ and $3$.
$$(m - 5)(m + 3) = 0$$

Step5: Solve for m

Set each factor equal to 0:
$m - 5 = 0 \implies m=5$
$m + 3 = 0 \implies m=-3$
Check that these do not violate non-permissible values ($m
eq 0$), which they do not.

Answer:

Non-permissible value: $m
eq 0$
Solutions: $m=5$ and $m=-3$