Solve for x

1. x + 4 over x - 2 = 0

There is only one way to reach 0 by division.

⌧ not N/N because N/N = 1, not 0

⌧ not N/0 because N/0 = undefined, not 0

☑ Yes 0/N = 0

So if the answer to a division problem is 0, then the numerator must have been 0 and the denominator could have been anything.

In the case of this problem,

x + 4

--------- = 0

x - 2

For the numerator to end up as 0, x obviously has to be - 4 ← answer

Answer:

x = - 4

------------------

You can get that same answer by math too -- just by mechanically solving for x

Solve for x

x + 4

--------- = 0

x - 2

1) Clear the fraction by multiplying both sides by x-2 and letting the denominator cancel. After you have multiplied and canceled, you get this:

x + 4 = 0

2) Subtract 4 from both sides to isolate x

x = - 4 ← same answer

Check

Sub in -4 in the place of x

x + 4

--------- = 0

x - 2

-4 + 4

---------- should equal 0

x - 2

0 over x - 2 does equal 0 ✓

-----------------------------

2. x² - 4 over x - 2 = 2

Solving this problem depends on recognizing that (x² - 4) is a Difference of Two Perfect Squares.

Difference of Two Squares problems are factored by memorization

x² - y²

(x + y)(x - y)

Solve for x

x² - 4

--------- = 2

x - 2

1) Factor the numerator as the Difference of Two Squares to find factors that can cancel with the denominator

(x - 2)( x + 2)

------------------- = 2

(x - 2)

2) Cancel (x - 2) from the top and the bottom. After you cancel, you get this:

x + 2 = 2

3) Subtract 2 from both sides to isolate x

x = 0 ← answer

Check

Sub in 0 in the place of x

x² - 4

--------- = 2

x - 2

0² - 4

--------- should equal 2

0 - 2

- 4 over - 2 does equal 2 ✓

------------------------- -----------------------

3. Solve for x

x² + x - 2

-------------- = 3

x - 3

1) Clear the fraction by multiplying both sides by (x - 3) and letting the denominator cancel. After you have multiplied and canceled, you have this:

x² + x - 2 = 3( x - 3)

- same as -

x² + x - 2 = 3x - 9

2) Subtract 3x from both sides to collect all the x terms on the same side

x² - 2x - 2 = - 9

3) Add 9 to both sides to set the equation equal to 0

x² - 2x + 7 = 0

This doesn't factor

You can try the quadratic formula, I guess.

The left side cancels

x² + x - 2 = 3x - 9

(x + 2)(x - 1) = 3(x - 3)

But this doesn't get you anywhere either