Chapter 14, Problem 1IR

Determine whether each o f the following is an equation or an expression. If it is an equation, solve it for its variable. If it is an expression, perform the indicated operation.

1. $\frac{1}{x}+\frac{2}{3}$

To determine

Check whether $\frac{1}{x}+\frac{2}{3}$ is an equation or an expression

Solve if it’s an equation otherwise perform the indicated operation of expression

Answer to Problem 1IR

  $\frac{1}{x}+\frac{2}{3}$ is an expression and $\frac{1}{x}+\frac{2}{3}=\frac{3+2x}{3x}$

Explanation of Solution

Given:

  $\frac{1}{x}+\frac{2}{3}$

Concept Used:

• To check if the given problem is an equation or an equation we have to look for “ equals to ” if there is an = sign in problem then it is an equation, otherwise it’s an expression
• To add/subtract rational expressions with unlike denominator
1. find LCD(least common divisor) of denominators
2. convert each rational expressions as an equivalent fraction whose denominator is the LCD
3. Add/subtract the like rational expressions
• To simplify a given expression, we have to find the common factor of the numerator and denominator and then divide them by the common factor.

Calculation:

To check whether $\frac{1}{x}+\frac{2}{3}$ is an equation or an expression we have to look for “ equals to ” if there is an = sign in problem then it is an equation, otherwise it’s an expression, as there is no equals to sign in $\frac{1}{x}+\frac{2}{3}$ hence it’s an expression

Now, to perform the operation in $\frac{1}{x}+\frac{2}{3}$ which is addition, first see as the LCD of denominators in given rational expression ( x and 3) is 3x, thus write each rational expression as an equivalent rational expression whose denominator is 3x , thus

  $\frac{1}{x}\cdot \frac{3}{3}=\frac{3}{3x},\frac{2}{3}\cdot \frac{x}{x}=\frac{2x}{3x}$

Now, addition of like rational expression can be done as

  $\begin{array}{c}\frac{1}{x}+\frac{2}{3}=\frac{3}{3x}+\frac{2x}{3x}\\ =\frac{3+2x}{3x}\end{array}$

Thus, $\frac{1}{x}+\frac{2}{3}=\frac{3+2x}{3x}$