1.
Question 1. For which value(s) of \(q\) is \(\frac{12q-14}{q^2-64}\) not defined?
Step 1: A rational expression is not defined when its denominator equals zero. Step 2: Set the denominator equal to zero: \(q^2-64=0\). Step 3: Factor the difference of squares: \((q-8)(q+8)=0\). Step 4: Solve each factor: \(q=8\) or \(q=-8\). Answer: \(8, -8\)
2.
Question 2. Simplify \(\frac{(x^2-25)(x^2-9)}{x^2-6x-27}\).
Step 1: Factor the numerator: \(x^2-25=(x-5)(x+5)\) and \(x^2-9=(x-3)(x+3)\). Step 2: Factor the denominator: \(x^2-6x-27=(x-9)(x+3)\). Step 3: Substitute the factors: \(\frac{(x-5)(x+5)(x-3)(x+3)}{(x-9)(x+3)}\). Step 4: Cancel the common factor \((x+3)\). Step 5: The expression becomes \(\frac{(x^2-25)(x-3)}{x-9}\). Step 6: Expand the numerator: \((x^2-25)(x-3)=x^3-3x^2-25x+75\). Answer: \(\frac{x^3-3x^2-25x+75}{x-9}\)
3.
Question 3. Simplify \(\frac{2}{x-1}-\frac{2}{x^2+2x+1}\).
Step 1: Factor the second denominator: \(x^2+2x+1=(x+1)^2\). Step 2: The common denominator is \((x-1)(x+1)^2\). Step 3: Rewrite the first fraction: \(\frac{2}{x-1}=\frac{2(x+1)^2}{(x-1)(x+1)^2}\). Step 4: Rewrite the second fraction: \(\frac{2}{(x+1)^2}=\frac{2(x-1)}{(x-1)(x+1)^2}\). Step 5: Subtract the numerators: \(2(x+1)^2-2(x-1)\). Step 6: Expand and simplify: \(2(x^2+2x+1)-2x+2=2x^2+2x+4=2(x^2+x+2)\). Answer: \(\frac{2(x^2+x+2)}{(x+1)^2(x-1)}\)
4.
Question 4. Simplify \(\frac{6x^2-8}{x-3y}\times\frac{36y^2-4x^2}{9x^2-12}\).
Step 1: Factor each expression: \(6x^2-8=2(3x^2-4)\). Step 2: Factor \(36y^2-4x^2\) as a difference of squares: \(36y^2-4x^2=4(9y^2-x^2)=-4(x-3y)(x+3y)\). Step 3: Factor the denominator: \(9x^2-12=3(3x^2-4)\). Step 4: Substitute: \(\frac{2(3x^2-4)}{x-3y}\times\frac{-4(x-3y)(x+3y)}{3(3x^2-4)}\). Step 5: Cancel \((3x^2-4)\) and \((x-3y)\). Step 6: Multiply the remaining constants: \(\frac{2(-4)(x+3y)}{3}=-\frac{8(x+3y)}{3}\). Answer: \(-\frac{8(3y+x)}{3}\)
5.
Question 5. Simplify the rational expression \(\frac{a^2-7a-18}{9-a}\div\frac{a-9}{3-a}\).
Step 1: Change division to multiplication by the reciprocal: \(\frac{a^2-7a-18}{9-a}\times\frac{3-a}{a-9}\). Step 2: Factor the numerator: \(a^2-7a-18=(a-9)(a+2)\). Step 3: Substitute: \(\frac{(a-9)(a+2)}{9-a}\times\frac{3-a}{a-9}\). Step 4: Cancel \((a-9)\). Step 5: The expression becomes \(\frac{(a+2)(3-a)}{9-a}\). Step 6: Expand the numerator: \((a+2)(3-a)=3a-a^2+6-2a=-a^2+a+6\). Answer: \(\frac{-a^2+a+6}{9-a}\)
6.
Question 6. The area of a rectangular box is \(24x^3-66x^2+15x\) square units. The width of this box is \((6x^2-15x)\) units. Write and simplify an expression for the length of the rectangle.
Step 1: Use \(\text{Area}=\text{length}\times\text{width}\). Step 2: So \(\text{length}=\frac{24x^3-66x^2+15x}{6x^2-15x}\). Step 3: Factor the numerator: \(24x^3-66x^2+15x=3x(8x^2-22x+5)\). Step 4: Factor the trinomial: \(8x^2-22x+5=(4x-1)(2x-5)\). Step 5: Factor the denominator: \(6x^2-15x=3x(2x-5)\). Step 6: Cancel the common factors \(3x\) and \((2x-5)\). Answer: \((4x-1)\) units
7.
Question 7. Solve for \(x\): \(\frac{x}{7}+\frac{4}{x}=\frac{x+4}{7}\).
Step 1: Identify the restriction: \(x\neq0\). Step 2: Multiply every term by the common denominator \(7x\). Step 3: \(7x\cdot\frac{x}{7}+7x\cdot\frac{4}{x}=7x\cdot\frac{x+4}{7}\). Step 4: Simplify: \(x^2+28=x(x+4)\). Step 5: Expand: \(x^2+28=x^2+4x\). Step 6: Subtract \(x^2\) from both sides: \(28=4x\). Step 7: Solve: \(x=7\). Answer: \(7\)
8.
Question 8. Solve \(\frac{1}{2x-2}=\frac{4}{5x-5}\).
Step 1: Factor the denominators: \(2x-2=2(x-1)\) and \(5x-5=5(x-1)\). Step 2: The restriction is \(x\neq1\). Step 3: Cross multiply: \(1\cdot(5x-5)=4(2x-2)\). Step 4: Simplify: \(5x-5=8x-8\). Step 5: Solve: \(3=3x\), so \(x=1\). Step 6: Since \(x=1\) is not allowed, there is no real solution. Answer: No real solution
9.
Question 9. Solve the rational equation \(x+\frac{9}{x+4}-6=0\). The product of the two solutions is?
Step 1: Start with \(x+\frac{9}{x+4}-6=0\). Step 2: Multiply every term by \(x+4\): \(x(x+4)+9-6(x+4)=0\). Step 3: Expand: \(x^2+4x+9-6x-24=0\). Step 4: Combine like terms: \(x^2-2x-15=0\). Step 5: Factor: \((x-5)(x+3)=0\). Step 6: The two solutions are \(x=5\) and \(x=-3\). Step 7: Their product is \(5(-3)=-15\). Answer: \(-15\)
10.
Question 10. On average, Sherry can travel four times as fast on cross-country skis as she can with snowshoes. To travel \(24\) km, she needs \(6\) h more if she is snowshoeing than if she is skiing. What is her average speed on cross-country skis?
Step 1: Let Sherry's snowshoeing speed be \(r\) km/h. Step 2: Her skiing speed is four times as fast, so it is \(4r\) km/h. Step 3: Time equals distance divided by speed. Snowshoeing time is \(\frac{24}{r}\), and skiing time is \(\frac{24}{4r}\). Step 4: Snowshoeing takes 6 hours longer, so \(\frac{24}{r}-\frac{24}{4r}=6\). Step 5: Simplify \(\frac{24}{4r}=\frac{6}{r}\). Step 6: Then \(\frac{24}{r}-\frac{6}{r}=6\), so \(\frac{18}{r}=6\). Step 7: Solve: \(18=6r\), so \(r=3\). Step 8: Skiing speed is \(4r=4(3)=12\). Answer: \(12\) km/h
11.
Question 11. Two consecutive even whole numbers are selected. The difference between the reciprocals of the two numbers is \(\frac{1}{24}\). Determine the two numbers.
Step 1: Let the smaller even whole number be \(x\). The next consecutive even whole number is \(x+2\). Step 2: Set up the equation: \(\frac{1}{x}-\frac{1}{x+2}=\frac{1}{24}\). Step 3: Combine the left side: \(\frac{x+2-x}{x(x+2)}=\frac{1}{24}\). Step 4: Simplify: \(\frac{2}{x(x+2)}=\frac{1}{24}\). Step 5: Cross multiply: \(48=x(x+2)\). Step 6: Expand: \(x^2+2x-48=0\). Step 7: Factor: \((x+8)(x-6)=0\). Step 8: Since the numbers are whole numbers, \(x=6\), and the next even number is \(8\). Answer: \(6\) and \(8\)
12.
Question 12. Small tug boats pull cruise ships up and down rivers to prevent them from grounding. Cruise ships anchor \(22\) km away from the river port. A tug boat can travel \(20\) km downstream in the same time it takes it to travel \(10\) km upstream. If the speed of the current is \(5\) km/h, calculate the time it takes for the tug boats to travel upstream from the cruise ship to the river port.
Step 1: Let the boat speed in still water be \(v\) km/h. Step 2: The downstream speed is \(v+5\), and the upstream speed is \(v-5\). Step 3: The time for \(20\) km downstream equals the time for \(10\) km upstream, so \(\frac{20}{v+5}=\frac{10}{v-5}\). Step 4: Cross multiply: \(20(v-5)=10(v+5)\). Step 5: Expand: \(20v-100=10v+50\). Step 6: Solve: \(10v=150\), so \(v=15\). Step 7: The upstream speed is \(15-5=10\) km/h. Step 8: The cruise ship is \(22\) km from the port, so upstream time is \(\frac{22}{10}=2.2\) hours. Step 9: Convert to minutes: \(2.2\times60=132\) minutes. Answer: \(132\) minutes
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