Welcome to our lesson on simplifying algebraic expressions with fractions. This process involves several key steps. First, we find a common denominator for all fractions in the expression. Then, we rewrite each fraction using this common denominator. Next, we combine the numerators while keeping the common denominator. After that, we simplify the resulting expression by combining like terms. Finally, we cancel any common factors between the numerator and denominator. Let's see how this works with an example.
Let's work through a specific example: simplifying the expression x over x plus 1, plus 2 over x minus 1. First, we need to find a common denominator. Since we have x plus 1 and x minus 1 as denominators, our common denominator will be their product: (x plus 1) times (x minus 1). Next, we rewrite each fraction with this common denominator. For the first fraction, we multiply both numerator and denominator by x minus 1. For the second fraction, we multiply by x plus 1. This gives us x times x minus 1 over (x plus 1) times (x minus 1), plus 2 times x plus 1 over (x minus 1) times (x plus 1).
Now we continue with our simplification. In step 3, we expand the numerators. The first numerator x times (x minus 1) becomes x squared minus x. The second numerator 2 times (x plus 1) becomes 2x plus 2. In step 4, we combine these expressions over the common denominator: (x squared minus x) plus (2x plus 2) all over (x plus 1)(x minus 1). When we combine like terms in the numerator, we get x squared plus x plus 2 over (x plus 1)(x minus 1). This is our simplified expression. Note that we can also write the denominator as x squared minus 1. And we must remember the domain restriction: x cannot equal 1 or negative 1, as these values would make the denominator zero.
Let's look at another example: 3 over x minus 2, minus 4 over x plus 3. We follow the same steps as before. First, we find the common denominator, which is (x minus 2) times (x plus 3). Next, we rewrite each fraction with this common denominator. For the first fraction, we multiply both numerator and denominator by x plus 3. For the second fraction, we multiply by x minus 2. This gives us 3 times x plus 3 over (x minus 2)(x plus 3), minus 4 times x minus 2 over (x plus 3)(x minus 2). When we expand the numerators, we get 3x plus 9 over (x minus 2)(x plus 3), minus 4x minus 8 over (x plus 3)(x minus 2).
To summarize what we've learned about simplifying algebraic expressions with fractions: First, always find a common denominator for all fractions in the expression. Second, rewrite each fraction using this common denominator by multiplying both numerator and denominator by the appropriate factors. Third, combine the numerators while keeping the common denominator - remember to be careful with signs when subtracting. Fourth, simplify the resulting expression by combining like terms, factoring when possible, and canceling common factors between numerator and denominator. Finally, always specify domain restrictions where the denominator would equal zero. These steps will help you simplify any algebraic expression containing fractions.