1.
Item 5. Simplify \(\frac{5x}{2}\times\frac{24x}{75}\).
Solution:\(\frac{5x}{2}\times\frac{24x}{75}=\frac{120x^2}{150}\).Simplify by dividing numerator and denominator by 30:\(=\frac{4x^2}{5}\).There is no variable in any denominator, so there are no restrictions on x.Answer: D
2.
Item 6. Simplify \(\frac{3x-9}{5x-15}\times\frac{2x-3}{-4x^2+9}\).
Solution:Factor everything:\(3x-9=3(x-3)\)\(5x-15=5(x-3)\)\(-4x^2+9=-(2x-3)(2x+3)\)Then:\(\frac{3(x-3)}{5(x-3)}\times\frac{2x-3}{-(2x-3)(2x+3)}\)Cancel common factors:\(=\frac{3}{5}\times\frac{1}{-(2x+3)}\)\(=\frac{-3}{5(2x+3)}\).Answer: D
3.
Item 3. What would replace the box to make \(\frac{12}{5x}\) equivalent to \(\frac{口}{25x}\)?
Solution:To change \(5x\) into \(25x\), multiply by 5.Multiply the numerator by the same factor:\(12\times5=60\).Answer: A
4.
Item 1. Which of the following are NOT rational expressions? A. \(\frac{5x}{3}\),B. \(\frac{1}{6}\),C. \(\frac{1-\sqrt{a}}{1+\sqrt{a}}\),D. \(2x-4\)
Solution:A and B are rational expressions because they are ratios of polynomials.D is also a polynomial, therefore a rational expression.C contains \(\sqrt{a}\), so the variable is inside a radical and it is NOT a rational expression.Answer: A
5.
Item 2. Determine the non-permissible values of \(\frac{1}{x^2+4}\).
Solution:The denominator is \(x^2+4\).For real numbers, \(x^2\ge0\), therefore \(x^2+4>0\) always.The denominator can never equal zero.There are no restrictions on x.Answer: C
6.
Item 4. Simplify \(\frac{x^2+x-12}{(x-3)^2}\) and state restrictions.
Solution:Factor numerator:\(x^2+x-12=(x+4)(x-3)\).\(\frac{(x+4)(x-3)}{(x-3)^2}=\frac{x+4}{x-3}\).Restriction comes from original denominator: \(x-3\neq0\).Therefore \(x\neq3\).Answer: D
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