1.
Item 1. Simplify and state the non-permissible value(s). \(\frac{8x-80}{24y}\div\frac{x-10}{6y}\)
Solution:\(\frac{8x-80}{24y}\div\frac{x-10}{6y}=\frac{8(x-10)}{24y}\times\frac{6y}{x-10}\).Cancel \((x-10)\) and \(y\):\(=\frac{8\cdot6}{24}=2\).Restrictions from original expression: \(y\neq0\) and \(x\neq10\).Answer: C
2.
Item 6. Solve and state restrictions. \(\frac5{5t+8}=\frac7{t-4}\)
Solution:Cross multiply:\(5(t-4)=7(5t+8)\).\(5t-20=35t+56\).\(-30t=76\).\(t=-\frac{76}{30}=-\frac{38}{15}\).Restrictions: \(5t+8\neq0\Rightarrow t\neq-\frac85\), and \(t\neq4\).Answer: A
3.
Item 4. Simplify \(\frac7{6x^2-8x-8}+\frac{8-x}{15x^2+13x+2}-\frac9{10x^2-18x-4}\)
Solution:Factor denominators:\(6x^2-8x-8=2(3x+2)(x-2)\)\(15x^2+13x+2=(3x+2)(5x+1)\)\(10x^2-18x-4=2(5x+1)(x-2)\).Use LCD \(2(3x+2)(x-2)(5x+1)\).Combine numerators and simplify.Result: \(\frac{-2x^2+28x-43}{2(3x+2)(x-2)(5x+1)}\).Answer: C
4.
Item 2. Simplify \(\frac{x^2-64}{x^2-49}\div\frac{2x^2+17x+8}{8x^2+60x+28}\)
Solution:Factor:\(x^2-64=(x-8)(x+8)\), \(x^2-49=(x-7)(x+7)\).\(2x^2+17x+8=(2x+1)(x+8)\).\(8x^2+60x+28=4(2x+1)(x+7)\).Multiply by reciprocal and cancel common factors.Result: \(\frac{4(x-8)}{x-7}\).Answer: D
5.
Item 3. Simplify \(\frac{x+1}{2x^2-2}+\frac{2x-6}{4x^2-8x+4}\)
Solution:Factor denominators:\(2x^2-2=2(x-1)(x+1)\), \(4x^2-8x+4=4(x-1)^2\).Find LCD \(=4(x-1)^2(x+1)\).Simplify and combine numerators.Final result: \(\frac{x-2}{(x-1)^2}\).Answer: D
6.
Item 5. Solve and state restrictions. \(\frac5{7x}-\frac83=43\)
Solution:LCD = \(21x\).Multiply through:\(15-56x=903x\).\(-959x=-15\).\(x=\frac{15}{959}\).Restriction: \(x\neq0\).Answer: D
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