Practice Exercises 2.1
Solve the following rational equations. Check for extraneous solutions.
1. x - = -5
6.
6
5
3
+
3x
4
х
5x+6
5
2a
2. 3x =
7.2
2x+3
a
a+1
x²-1
х
8
3.
278-9
x2 +8x+7
11
+
x+7
8.
x-3
x-3
x2
4x-2
1
2
4.
+
9.
X
6
+
X-8
x-2
-2-x-12 x+3
x2-10-16
X-4
3c
9
2
1
4
5.
+
10.
10
c²-4
C-2
C+2
XX2

A rational equation is an equation containing at least one rational expression. Rational expressions typically contain a variable in the denominator. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and checking our solutions.

Solve rational equations by clearing the fractions by multiplying both sides of the equation by the least common denominator (LCD).

Step-by-step explanation:

After multiplying both sides of the previous example by the LCD, we were left with a linear equation to solve. This is not always the case; sometimes we will be left with a quadratic equation.

After distributing and dividing out the common factors, a quadratic equation remains. To solve it, rewrite it in standard form, factor, and then set each factor equal to 0.

Up to this point, all of the possible solutions have solved the original equation. However, this may not always be the case. Multiplying both sides of an equation by variable factors may lead to extraneous solutions, which are solutions that do not solve the original equation. A complete list of steps for solving a rational equation is outlined in the following example.