1.
Question 1. For which value(s) of \(n\) is \(\frac{5n-4}{n^2-4}\) not defined?
Step 1: A rational expression is not defined when its denominator is equal to zero.\nStep 2: Set the denominator equal to zero: \(n^2-4=0\).\nStep 3: Factor the difference of squares: \((n-2)(n+2)=0\).\nStep 4: Solve each factor: \(n-2=0\) gives \(n=2\), and \(n+2=0\) gives \(n=-2\).\nAnswer: \(2, -2\)
2.
Question 2. Simplify: \(\frac{8t^2-128}{4t^2+36t+80}\).
Step 1: Factor the numerator: \(8t^2-128=8(t^2-16)=8(t-4)(t+4)\).\nStep 2: Factor the denominator: \(4t^2+36t+80=4(t^2+9t+20)=4(t+4)(t+5)\).\nStep 3: Write the expression as \(\frac{8(t-4)(t+4)}{4(t+4)(t+5)}\).\nStep 4: Cancel the common factor \((t+4)\), and simplify \(\frac{8}{4}=2\).\nAnswer: \(\frac{2(t-4)}{t+5}\)
3.
Question 3. Simplify: \(\frac{(x^2-49)(x^2-81)}{x^2+15x+56}\).
Step 1: Factor \(x^2-49\) as a difference of squares: \(x^2-49=(x-7)(x+7)\).\nStep 2: Factor the denominator: \(x^2+15x+56=(x+7)(x+8)\).\nStep 3: Substitute the factors into the expression: \(\frac{(x-7)(x+7)(x^2-81)}{(x+7)(x+8)}\).\nStep 4: Cancel the common factor \((x+7)\).\nAnswer: \(\frac{(x-7)(x^2-81)}{x+8}\)
4.
Question 4. Simplify: \(\frac{6m}{3}-\frac{3m}{6}+\frac{m}{3}\).
Step 1: Simplify each term where possible: \(\frac{6m}{3}=2m\) and \(\frac{3m}{6}=\frac{m}{2}\).\nStep 2: The expression becomes \(2m-\frac{m}{2}+\frac{m}{3}\).\nStep 3: Use the common denominator \(6\): \(2m=\frac{12m}{6}\), \(\frac{m}{2}=\frac{3m}{6}\), and \(\frac{m}{3}=\frac{2m}{6}\).\nStep 4: Combine the numerators: \(\frac{12m-3m+2m}{6}=\frac{11m}{6}\).\nAnswer: \(\frac{11m}{6}\)
5.
Question 5. Simplify: \(\frac{7x}{2x+4}-\frac{x-2}{3x+6}\).
Step 1: Factor the denominators: \(2x+4=2(x+2)\) and \(3x+6=3(x+2)\).\nStep 2: The lowest common denominator is \(6(x+2)\).\nStep 3: Rewrite each fraction: \(\frac{7x}{2(x+2)}=\frac{21x}{6(x+2)}\) and \(\frac{x-2}{3(x+2)}=\frac{2(x-2)}{6(x+2)}\).\nStep 4: Subtract the numerators: \(21x-2(x-2)=21x-2x+4=19x+4\).\nStep 5: Write the simplified expression: \(\frac{19x+4}{6(x+2)}=\frac{19x+4}{6x+12}\).\nAnswer: \(\frac{19x+4}{6x+12}\)
6.
Question 6. Simplify: \(\frac{2}{x-1}-\frac{2}{x^2+2x+1}\).
Step 1: Factor the second denominator: \(x^2+2x+1=(x+1)^2\).\nStep 2: The lowest common denominator is \((x-1)(x+1)^2\).\nStep 3: Rewrite the first fraction: \(\frac{2}{x-1}=\frac{2(x+1)^2}{(x-1)(x+1)^2}\).\nStep 4: Rewrite the second fraction: \(\frac{2}{(x+1)^2}=\frac{2(x-1)}{(x-1)(x+1)^2}\).\nStep 5: Subtract the numerators: \(2(x+1)^2-2(x-1)\).\nStep 6: Expand and simplify: \(2(x^2+2x+1)-2x+2=2x^2+2x+4=2(x^2+x+2)\).\nAnswer: \(\frac{2(x^2+x+2)}{(x+1)^2(x-1)}\)
7.
Question 7. Simplify: \(\frac{4a}{2b}\times\frac{9b^2}{6a}\).
Step 1: Multiply the numerators and denominators: \(\frac{4a}{2b}\times\frac{9b^2}{6a}=\frac{36ab^2}{12ab}\).\nStep 2: Simplify the coefficient: \(\frac{36}{12}=3\).\nStep 3: Cancel the common factor \(a\).\nStep 4: Simplify \(\frac{b^2}{b}=b\).\nAnswer: \(3b\)
8.
Question 8. Simplify: \(\frac{6x^2-8}{2x-y}\times\frac{4x^2-y^2}{9x^2-12}\).
Step 1: Factor each expression where possible: \(6x^2-8=2(3x^2-4)\).\nStep 2: Factor the difference of squares: \(4x^2-y^2=(2x-y)(2x+y)\).\nStep 3: Factor the other denominator: \(9x^2-12=3(3x^2-4)\).\nStep 4: Substitute the factors: \(\frac{2(3x^2-4)}{2x-y}\times\frac{(2x-y)(2x+y)}{3(3x^2-4)}\).\nStep 5: Cancel \((3x^2-4)\) and \((2x-y)\).\nAnswer: \(\frac{2(2x+y)}{3}\)
9.
Question 9. 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: \(\frac{a^2-7a-18}{9-a}\times\frac{3-a}{a-9}\).\nStep 2: Factor the numerator: \(a^2-7a-18=(a-9)(a+2)\).\nStep 3: Substitute the factorization: \(\frac{(a-9)(a+2)}{9-a}\times\frac{3-a}{a-9}\).\nStep 4: Cancel the common factor \((a-9)\).\nStep 5: The expression becomes \(\frac{(a+2)(3-a)}{9-a}\).\nStep 6: Expand the numerator: \((a+2)(3-a)=3a-a^2+6-2a=-a^2+a+6\).\nAnswer: \(\frac{-a^2+a+6}{9-a}\)
10.
Question 10. 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 the formula \(\text{Area}=\text{length}\times\text{width}\).\nStep 2: Solve for length: \(\text{length}=\frac{24x^3-66x^2+15x}{6x^2-15x}\).\nStep 3: Factor the numerator: \(24x^3-66x^2+15x=3x(8x^2-22x+5)=3x(4x-1)(2x-5)\).\nStep 4: Factor the denominator: \(6x^2-15x=3x(2x-5)\).\nStep 5: Cancel the common factors \(3x\) and \((2x-5)\).\nAnswer: \((4x-1)\) units
11.
Question 11. Solve for \(x\): \(\frac{7}{x}-\frac{4}{6x}=8\).
Step 1: Simplify the second fraction: \(\frac{4}{6x}=\frac{2}{3x}\).\nStep 2: Rewrite the equation: \(\frac{7}{x}-\frac{2}{3x}=8\).\nStep 3: Use the common denominator \(3x\): \(\frac{21}{3x}-\frac{2}{3x}=8\).\nStep 4: Combine the fractions: \(\frac{19}{3x}=8\).\nStep 5: Multiply both sides by \(3x\): \(19=24x\).\nStep 6: Divide by \(24\): \(x=\frac{19}{24}\).\nAnswer: \(\frac{19}{24}\)
12.
Question 12. Solve for \(x\): \(\frac{x}{7}+\frac{4}{x}=\frac{x+4}{7}\).
Step 1: Identify the restrictions: \(x\neq 0\).\nStep 2: Multiply every term by the lowest common denominator \(7x\).\nStep 3: \(7x\cdot\frac{x}{7}+7x\cdot\frac{4}{x}=7x\cdot\frac{x+4}{7}\).\nStep 4: Simplify: \(x^2+28=x(x+4)\).\nStep 5: Expand the right side: \(x^2+28=x^2+4x\).\nStep 6: Subtract \(x^2\) from both sides: \(28=4x\).\nStep 7: Divide by \(4\): \(x=7\).\nAnswer: \(7\)
13.
Question 13. Solve \(\frac{1}{2x-2}=\frac{4}{5x-5}\).
Step 1: Factor the denominators: \(2x-2=2(x-1)\) and \(5x-5=5(x-1)\).\nStep 2: The restriction is \(x\neq 1\).\nStep 3: Multiply both sides by \(10(x-1)\), the lowest common denominator.\nStep 4: \(10(x-1)\cdot\frac{1}{2(x-1)}=10(x-1)\cdot\frac{4}{5(x-1)}\).\nStep 5: Simplify both sides: \(5=8\).\nStep 6: Since \(5=8\) is false, there is no value of \(x\) that satisfies the equation.\nAnswer: No real solution
14.
Question 14. Solve: \(x+\frac{9}{x+4}=6\).
Step 1: Identify the restriction: \(x\neq -4\).\nStep 2: Move \(6\) to the left side: \(x-6+\frac{9}{x+4}=0\).\nStep 3: Multiply both sides by \(x+4\): \((x-6)(x+4)+9=0\).\nStep 4: Expand: \(x^2+4x-6x-24+9=0\).\nStep 5: Simplify: \(x^2-2x-15=0\).\nStep 6: Factor: \((x-5)(x+3)=0\).\nStep 7: Solve each factor: \(x=5\) or \(x=-3\).\nStep 8: Both values are allowed because neither equals \(-4\).\nAnswer: \(5, -3\)
15.
Question 15. On average, Jaimie can bike five times as fast as he can run. To travel \(25\) km, he needs \(2\) h more if he is running than if he is biking. What is his average running speed?
Step 1: Let Jaimie's running speed be \(r\) km/h.\nStep 2: His biking speed is five times as fast, so it is \(5r\) km/h.\nStep 3: Time equals distance divided by speed. Running time is \(\frac{25}{r}\), and biking time is \(\frac{25}{5r}\).\nStep 4: Running takes \(2\) hours longer, so \(\frac{25}{r}-\frac{25}{5r}=2\).\nStep 5: Simplify \(\frac{25}{5r}=\frac{5}{r}\), so \(\frac{25}{r}-\frac{5}{r}=2\).\nStep 6: Combine: \(\frac{20}{r}=2\).\nStep 7: Multiply by \(r\): \(20=2r\).\nStep 8: Divide by \(2\): \(r=10\).\nAnswer: \(10\) km/h
16.
Question 16. Two consecutive even whole numbers are selected. The difference between the reciprocals of the two numbers is \(\frac{1}{40}\). Determine the two numbers.
Step 1: Let the smaller even whole number be \(n\). The next consecutive even whole number is \(n+2\).\nStep 2: Set up the equation for the difference of the reciprocals: \(\frac{1}{n}-\frac{1}{n+2}=\frac{1}{40}\).\nStep 3: Combine the left side: \(\frac{(n+2)-n}{n(n+2)}=\frac{1}{40}\).\nStep 4: Simplify the numerator: \(\frac{2}{n(n+2)}=\frac{1}{40}\).\nStep 5: Cross multiply: \(80=n(n+2)\).\nStep 6: Expand: \(n^2+2n=80\).\nStep 7: Rearrange: \(n^2+2n-80=0\).\nStep 8: Factor: \((n+10)(n-8)=0\).\nStep 9: Since the numbers are whole numbers, choose \(n=8\). The next even number is \(10\).\nAnswer: \(8\) and \(10\)
17.
Question 17. South Central High Students’ Council is traveling to Calgary for the Students’ Council National Conference. From the travel budget allowed for the trip, they have two options. They can leave tonight by bus, or they can save two hours by leaving tomorrow morning and using the express train which travels \(20\) km/h faster than the bus. If the distance between South Central High and Calgary is \(800\) km, determine how long it will take to travel by bus, rounded to the nearest hour.
Step 1: Let the speed of the bus be \(v\) km/h.\nStep 2: The speed of the express train is \(v+20\) km/h.\nStep 3: The bus time is \(\frac{800}{v}\), and the train time is \(\frac{800}{v+20}\).\nStep 4: The train saves \(2\) hours, so \(\frac{800}{v}-\frac{800}{v+20}=2\).\nStep 5: Multiply both sides by \(v(v+20)\): \(800(v+20)-800v=2v(v+20)\).\nStep 6: Simplify the left side: \(16000=2v^2+40v\).\nStep 7: Divide by \(2\): \(8000=v^2+20v\).\nStep 8: Rearrange: \(v^2+20v-8000=0\).\nStep 9: Factor or solve: \((v-80)(v+100)=0\), so the positive speed is \(v=80\) km/h.\nStep 10: Find the bus time: \(\frac{800}{80}=10\).\nAnswer: \(10\) hours
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