For what value of [tex]$x$[/tex] is [tex]$\frac{3 x+2}{2 x+1}$[/tex] undefined?
A. [tex]$\frac{1}{2}$[/tex]
B. [tex]$-\frac{1}{2}$[/tex]
C. [tex]$-\frac{2}{3}$[/tex]
D. [tex]$\frac{2}{3}$[/tex]
Answer (1)
A rational expression is undefined when the denominator equals zero.
Set the denominator 2 x + 1 to zero: 2 x + 1 = 0 .
Solve for x : x = − 2 1 .
The expression is undefined when x = − 2 1 .
Explanation
Identifying the condition for undefined expression A rational expression is undefined when its denominator is equal to zero. In this case, the expression is 2 x + 1 3 x + 2 , so we need to find the value of x that makes the denominator, 2 x + 1 , equal to zero.
Setting up the equation To find the value of x that makes the denominator zero, we set up the equation 2 x + 1 = 0 .
Solving for x Now, we solve for x :
Subtract 1 from both sides of the equation: 2 x + 1 − 1 = 0 − 1 2 x = − 1 Divide both sides by 2: 2 2 x = 2 − 1 x = − 2 1
Finding the value of x Therefore, the expression 2 x + 1 3 x + 2 is undefined when x = − 2 1 .
Examples
Understanding when a rational function is undefined is crucial in many real-world applications, such as determining the domain of a function in physics or engineering. For example, if you're modeling the concentration of a substance over time with a rational function, knowing when the function is undefined helps you identify time points where the model breaks down or becomes physically impossible, such as negative time or infinite concentration. This ensures that the model remains valid and provides meaningful predictions within its applicable range.
