Add or subtract, as indicated. See Example 4. 1 1 ——— + ——— x + z x - z

Verified step by step guidance

1. Identify the common denominator for the fractions. In this case, the common denominator is $(x+z)(x-z)$.
2. Rewrite each fraction with the common denominator. The first fraction becomes $\frac{1}{x+z}=\frac{1\cdot (x-z)}{(x+z)(x-z)}$ and the second fraction becomes $\frac{1}{x-z}=\frac{1\cdot (x+z)}{(x+z)(x-z)}$.
3. Add the two fractions by combining the numerators over the common denominator: $\frac{1\cdot (x-z)+1\cdot (x+z)}{(x+z)(x-z)}$.
4. Simplify the numerator by combining like terms: $(x-z)+(x+z)$.
5. Further simplify the expression by canceling out terms or combining them, if possible.

Key Concepts

Fraction Addition and Subtraction

To add or subtract fractions, a common denominator is required. For the fractions given, the denominators are (x + z) and (x - z). The least common denominator (LCD) is the product of these two denominators, which allows for the fractions to be combined into a single fraction by adjusting the numerators accordingly.

Simplifying Fractions

After performing the addition or subtraction of fractions, the resulting fraction may need to be simplified. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number to reduce the fraction to its simplest form, making it easier to interpret and use.

Algebraic Manipulation

Algebraic manipulation is essential for rearranging and simplifying expressions. This includes distributing terms, combining like terms, and factoring when necessary. Understanding these techniques is crucial for effectively handling the algebraic expressions that arise during the addition or subtraction of fractions.