1.
Item 5. Solve and state restrictions. \(\frac5{7x}-\frac83=43\)
Solution:LCD = \(21x\).\(15-56x=903x\).\(-959x=-15\).\(x=\frac{15}{959}\).Restriction: \(x\neq0\).Answer: A
2.
Item 3. Simplify \(\frac{x+1}{2x^2-2}+\frac{2x-6}{4x^2-8x+4}\)
Solution:Factor denominators and use LCD \(2(x-1)^2(x+1)\).Combine fractions and simplify the numerator.Final result:\(\frac{x-2}{(x-1)^2}\).Answer: A
3.
Item 2. Simplify \(\frac{2x^2-7x-4}{30x^2+27x+6}\div\frac{x^2-16}{2x^2-10x-72}\)
Solution:Factor:\(2x^2-7x-4=(x-4)(2x+1)\)\(30x^2+27x+6=3(5x+2)(2x+1)\)\(x^2-16=(x-4)(x+4)\)\(2x^2-10x-72=2(x-9)(x+4)\).Multiply by the reciprocal and cancel common factors.Result: \(\frac{2(x-9)}{3(5x+2)}\).Answer: C
4.
Item 1. Simplify and state the non-permissible value. \(\frac{5x}{24y}\div\frac{18}{5}\)
Solution:\(\frac{5x}{24y}\div\frac{18}{5}=\frac{5x}{24y}\times\frac{5}{18}\).Multiply numerators and denominators:\(=\frac{25x}{432y}\).The denominator contains y, therefore \(y\neq0\).Answer: B
5.
Item 4. Simplify \(\frac{2x+1}{7x}+\frac{x-3}{7x}-\frac{x+5}{7x}\)
Solution:Common denominator is \(7x\).Combine numerators:\((2x+1)+(x-3)-(x+5)=2x-7\).Therefore:\(\frac{2x-7}{7x}\).Answer: D
6.
Item 6. Solve and state restrictions. \(\frac{x}{x-2}+\frac3{x-2}=\frac92\)
Solution:LCD = \(2(x-2)\).Multiply through:\(2x+6=9x-18\).\(-7x=-24\).\(x=\frac{24}{7}\).Restriction: \(x-2\neq0\Rightarrow x\neq2\).Answer: C
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