Multiply through by the common denominator. Note x ¹ 0 nor 2, since these values would make a denominator zero. If we get one of these in our solution we would throw it out.
3x2 + x - 2 = 3x2 - 6x
7x = 2
x = 2/7
More Notes for Equations
When solving rational equations watch out for extraneous solutions:
3x2 + x - 2 = 3x2 - 6x
7x = 2
x = 2/7
Now that we are solving equations a common mistake students make is to treat a rational expression as if it were an equation.
We can get a common denominator ofExample 2:and add this, but we can’t multiply both sides by the common denominator since we don’t have two sides!
2x2/5 + 7x1/5 + 3 = 0 You can let u = x1/5, then u2 = x2/5 .Example 3:
(2x1/5 + 1)(x1/5 + 3) = 0
x1/5 = -1/2 or x1/5 = -3
x = -1/32 or x = -243
Example 4:
See the book for a couple of example problems involving radicals. Remember anytime you raise both sides of an equation to an even power you need to check your solutions. Some of them may not work in the original problem, since for example: (-3)2 = 32 even though –3 ¹ 3.
Another type of problem that comes up in this section is absolute value equations. The main thing here is to remember to isolate the absolute value before getting rid of the absolute value.
Example: 3|2x –1| + 7 = 16
Isolate the absolute value: 3|2x - 1| = 9
|2x –1| = 3
Next, get rid of the radical: 2x - 1 = 3 or 2x - 1 = -3 since |3| = |-3| = 3
Solve the two resulting equations: 2x = 4 or 2x = -2
x = 2 or x = -1
