Question

for #1 - 7, a) simplify each expression to an equivalent expression in lowest terms. b) identify the value(s) of the variable that makes each expression undefined. 1. $\frac{5(x - 7)}{5x-35},x
eq0$ $\frac{x}{x - 7}$ 2. $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}$

Explanation:

Step1: Simplify the first - part of question 1a

Factor the numerator of $\frac{5x - 35}{x-7}$. We can factor out 5 from the numerator: $5x - 35=5(x - 7)$. Then $\frac{5x - 35}{x - 7}=\frac{5(x - 7)}{x - 7}$. Since $x
eq7$ (to avoid division by zero), we can cancel out the common factor $(x - 7)$.
$\frac{5(x - 7)}{x - 7}=5$

Step2: Find the undefined value of the first - part of question 1b

A rational expression $\frac{a}{b}$ is undefined when $b = 0$. For the expression $\frac{5x - 35}{x - 7}$, set the denominator $x-7 = 0$. Solving for $x$, we get $x = 7$.

Step3: Simplify the second - part of question 2

Factor the numerator and denominator of $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}$.
Factor $x^{2}+8x - 33=(x + 11)(x-3)$ and $x^{2}+5x - 24=(x + 8)(x - 3)$. Then $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}=\frac{(x + 11)(x - 3)}{(x + 8)(x - 3)}$. Since $x
eq3$ (to avoid division by zero), we can cancel out the common factor $(x - 3)$. So the simplified form is $\frac{x + 11}{x + 8}$.

Step4: Find the undefined value of the second - part of question 2

Set the denominator of $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}$ equal to zero. For the denominator $x^{2}+5x - 24=(x + 8)(x - 3)=0$. Solving $(x + 8)(x - 3)=0$ gives $x=-8$ or $x = 3$.

Answer:

1a. 5
1b. $x = 7$

1. Simplified form: $\frac{x + 11}{x + 8}$; Undefined values: $x=-8,x = 3$