Directions: You Must Show All Your Work For Each Problem
Directions You Must Show All Of Your Work For Each Problem All Answe
Directions: You must show all of your work for each problem. All answers should be written in simplest form. Always try to factor to see if it is possible to simplify! Only answers in simplest form will receive full credit.
Paper For Above instruction
Question 1: Solve for x: (x + 3)/6 = 5/(x + 2)
To solve for x in the equation \(\frac{x + 3}{6} = \frac{5}{x + 2}\), we cross-multiply to eliminate the denominators:
\( (x + 3)(x + 2) = 30 \)
Expanding the left side:
\( x^2 + 2x + 3x + 6 = 30 \)
Simplify the expression:
\( x^2 + 5x + 6 = 30 \)
Bring all terms to one side to set the equation to zero:
\( x^2 + 5x + 6 - 30 = 0 \)
\( x^2 + 5x - 24 = 0 \)
Factor the quadratic:
\( (x + 8)(x - 3) = 0 \)
Set each factor equal to zero:
\( x + 8 = 0 \Rightarrow x = -8 \)
\( x - 3 = 0 \Rightarrow x = 3 \)
Therefore, the solutions are \( x = -8 \) and \( x = 3 \).
Question 2: Find the quotient: (x² + 5x + 6) / (x² - 4)
First, factor numerator and denominator:
Numerator: \( x^2 + 5x + 6 = (x + 2)(x + 3) \)
Denominator: \( x^2 - 4 = (x - 2)(x + 2) \)
Express as a fraction:
\( \frac{(x + 2)(x + 3)}{(x - 2)(x + 2)} \)
Cancel the common factor \( x + 2 \):
\( \frac{\cancel{(x + 2)}(x + 3)}{(x - 2)\cancel{(x + 2)}} = \frac{x + 3}{x - 2} \)
Answer: \( \frac{x + 3}{x - 2} \)
Question 3: Find the sum: 4a + 6a²b + 2/3b + 4a + 6a²b + 2/3b
Combine like terms:
Sum of the \( 4a \) terms: \( 4a + 4a = 8a \)
Sum of the \( 6a^2b \) terms: \( 6a^2b + 6a^2b = 12a^2b \)
Sum of the \( 2/3b \) terms: \( 2/3b + 2/3b = 4/3b \)
Thus, the total sum is: \( 8a + 12a^2b + \frac{4}{3}b \)
Question 4: Solve for x: \(-4x + 3 + 2x = -8x + x + 3 + 2x\)
First, simplify both sides:
Left side: \(-4x + 2x + 3 = -2x + 3\)
Right side: \(-8x + x + 2x + 3 = (-8x + x + 2x) + 3 = (-5x + 2x) + 3 = -3x + 3\)
Set the simplified equations equal:
\( -2x + 3 = -3x + 3 \)
Subtract 3 from both sides:
\( -2x = -3x \)
Add 3x to both sides:
\( -2x + 3x = 0 \Rightarrow x = 0 \)
Solution: \( x = 0 \)
Question 5: Simplify: \( \frac{4x - 2x^3}{6x^3 + x + x - 2x^3 + 6x^3 + x + 2} \)
First, rewrite numerator and denominator carefully:
Numerator: \( 4x - 2x^3 \)
Denominator: \( 6x^3 + x + x - 2x^3 + 6x^3 + x + 2 \)
Simplify denominator step-by-step:
Combine like terms:
\( 6x^3 - 2x^3 + 6x^3 = (6 - 2 + 6) x^3 = 10x^3 \)
Sum the x terms: \( x + x + x = 3x \)
Constant term: 2
So, denominator: \( 10x^3 + 3x + 2 \)
The fraction becomes:
\( \frac{4x - 2x^3}{10x^3 + 3x + 2} \)
Factor numerator: \( 2x(2 - x^2) \), but since \( 2 - x^2 \) factors as \( ( \sqrt{2} - x)( \sqrt{2} + x) \), but for simplicity, we leave it factored as \( 2x(2 - x^2) \).
- Alternatively, no common factors exist here, so the expression is simplified as is.
Answer: \( \frac{4x - 2x^3}{10x^3 + 3x + 2} \)
Question 6: Find the sum: (x² + 3x – 2)/(x + 5) + (x – 2)/(x) + (4x + 12)/(x + 5)
Express all three fractions with common denominators:
First, rewrite each:
1. \( \frac{x^2 + 3x - 2}{x + 5} \)
2. \( \frac{x - 2}{x} \)
3. \( \frac{4x + 12}{x + 5} \)
Note that \( 4x + 12 = 4(x + 3) \). To combine these, the least common denominator (LCD) is \( x(x + 5) \). We rewrite each fraction with denominator \( x(x + 5) \):
- First term:
\( \frac{x^2 + 3x - 2}{x + 5} = \frac{(x^2 + 3x - 2) \times x}{x(x + 5)} = \frac{x(x^2 + 3x - 2)}{x(x + 5)} \)
- Second term:
\( \frac{x - 2}{x} = \frac{(x - 2)(x + 5)}{x(x + 5)} \)
- Third term:
\( \frac{4x + 12}{x + 5} = \frac{4(x + 3) \times x}{x(x + 5)} = \frac{4x(x + 3)}{x(x + 5)} \)
Now, write all as numerators over \( x(x + 5) \):
Numerator sum:
\( x(x^2 + 3x - 2) + (x - 2)(x + 5) + 4x(x + 3) \)
Compute each:
- \( x(x^2 + 3x - 2) = x^3 + 3x^2 - 2x \)
- \( (x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10 \)
- \( 4x(x + 3) = 4x^2 + 12x \)
Sum all:
\[ x^3 + 3x^2 - 2x + x^2 + 3x - 10 + 4x^2 + 12x \]
Combine like terms:
- Cubic term: \( x^3 \)
- Square terms: \( 3x^2 + x^2 + 4x^2 = 8x^2 \)
- x terms: \( -2x + 3x + 12x = 13x \)
- Constants: \( -10 \)
Therefore, total numerator:
\[
x^3 + 8x^2 + 13x - 10
\]
Final sum:
\[
\frac{x^3 + 8x^2 + 13x - 10}{x(x + 5)}
\]
Answer: \( \frac{x^3 + 8x^2 + 13x - 10}{x(x + 5)} \)
Question 7: Find the difference: \((-2x + 1)/(x^2 - 4) - (-3x - 1)/(x^2 - x + 1)\)
Express both fractions over a common denominator or leave in fractional form. The denominators are \( x^2 - 4 \) and \( x^2 - x + 1 \). We observe that \( x^2 - 4 = (x - 2)(x + 2) \). The second denominator does not factor over real numbers.
Express as a single difference:
\(\frac{-2x + 1}{(x - 2)(x + 2)} - \frac{-3x - 1}{x^2 - x + 1} \)
Find the least common denominator (LCD):
LCM of \((x - 2)(x + 2)\) and \( x^2 - x + 1 \). Since the quadratic does not factor over reals and the other factors are over reals, the LCD is their product:
\( (x - 2)(x + 2)(x^2 - x + 1) \)
Rewrite numerators to have this denominator:
First numerator:
\(\left(-2x + 1\right) \times (x^2 - x + 1)\)
Second numerator:
\(\left(-3x - 1\right) \times (x - 2)(x + 2)\)
Now, compute each numerator:
- Numerator 1:
\(-2x + 1\) times \( x^2 - x + 1 \):
\[
(-2x + 1)(x^2 - x + 1) = -2x \times x^2 + -2x \times -x + -2x \times 1 + 1 \times x^2 + 1 \times -x + 1 \times 1
\]
\[
= -2x^3 + 2x^2 -2x + x^2 - x + 1
\]
Simplify:
\[
-2x^3 + (2x^2 + x^2) + (-2x - x) + 1 = -2x^3 + 3x^2 - 3x + 1
\]
- Numerator 2:
\(-3x - 1\) times \((x - 2)(x + 2) = x^2 - 4\) (since \( (x - 2)(x + 2) = x^2 - 4 \)):
\[
(-3x - 1)(x^2 - 4) = -3x \times x^2 + -3x \times (-4) + -1 \times x^2 + -1 \times -4
\]
\[
= -3x^3 + 12x - x^2 + 4
\]
Now, combine the two numerators:
\[
\left( -2x^3 + 3x^2 - 3x + 1 \right) - \left( -3x^3 + 12x - x^2 + 4 \right)
\]
Distribute the minus:
\[
-2x^3 + 3x^2 - 3x + 1 + 3x^3 - 12x + x^2 - 4
\]
Combine like terms:
- \( x^3 \): \( -2x^3 + 3x^3 = x^3 \)
- \( x^2 \): \( 3x^2 + x^2 = 4x^2 \)
- \( x \): \( -3x - 12x = -15x \)
- constants: \( 1 - 4 = -3 \)
Final numerator:
\[
x^3 + 4x^2 - 15x - 3
\]
Thus, the entire difference is:
\[
\frac{ x^3 + 4x^2 - 15x - 3 }{ (x - 2)(x + 2)(x^2 - x + 1) }
\]
Answer: \(\frac{ x^3 + 4x^2 - 15x - 3 }{ (x - 2)(x + 2)(x^2 - x + 1) }\)
Question 8: Set up a proportion to solve: The Vikings led the Timberwolves by 19 points at halftime. If the Vikings scored 3 points for every 2 points the Timberwolves scored, what was the halftime score?
Let \( V \) be the points scored by the Vikings, and \( T \) be the points scored by the Timberwolves at halftime.
Given that:
- The lead is 19 points: \( V - T = 19 \)
- The scoring rate is proportionate: \( \frac{V}{T} = \frac{3}{2} \)
Set up the proportion: \( V / T = 3 / 2 \). Then, express \( V \) in terms of \( T \):
\( V = \frac{3}{2} T \)
Replace \( V \) in the lead equation:
\( \frac{3}{2} T - T = 19 \)
Simplify left side:
\( \left( \frac{3}{2} T - \frac{2}{2} T \right) = 19 \Rightarrow \frac{1}{2} T = 19 \)
Multiply both sides by 2:
\( T = 38 \)
Find \( V \):
\[
V = \frac{3}{2} \times 38 = 3 \times 19 = 57
\]
Therefore, halftime score:
- Vikings: 57 points
- Timberwolves: 38 points
Answer: Vikings 57, Timberwolves 38
References
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