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Which positive real number is equal to the quarter of its cube root?
Solution
Let \( x \) be the number to find and use the information above to write the equation
\[ x = \frac{1}{4} \sqrt[3]{x} \]
Multiply all terms of the equation by 4 and rewrite the equation as
\[ 4x = \sqrt[3]{x} \]
Cube both sides of the equation and simplify
\[ (4x)^3 = (\sqrt[3]{x})^3 \]
\[ 4^3 x^3 = x \]
Rewrite equation with zero on the right, factor and solve
\[ 4^3 x^3 - x = 0 \]
\[ x(4^3 x^2 - 1) = 0 \]
Solutions: \( x = 0 \) and \( 4^3 x^2 - 1 = 0 \), \( x^2 = \frac{1}{64} \) gives: \( x = \pm \frac{1}{8} \)
We are looking for a positive number, hence positive solution only
Answer: \( x = \frac{1}{8} \)
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If the points with coordinates \((a , b)\) and \((c , d)\) lie on the line with equation \(2y + 3x = 4\) and \(a - c = 3\), then what is the value of \(d - b\)?
Solution
Use the coordinates of the two points on the line to find the slope \( m \) of the line.
\[ m = \frac{d - b}{c - a} \]
The slope \( m \) can also be calculated using the equation of the line rewritten in the form \( y = m x + k \).
\[ 2y + 3x = 4 \implies 2y = -3x + 4 \implies y = -\frac{3}{2}x + 2 \]
The slope \( m \) is equal to \( -\frac{3}{2} \).
Equate the expression of the slope found above to \( -\frac{3}{2} \)
\[ -\frac{3}{2} = \frac{d - b}{c - a} \]
Use the above to find \( d - b \)
\[ d - b = -\frac{3}{2} (c - a) = -\frac{3}{2}(-(a - c)) = \frac{9}{2} \]
Answer: \( \frac{9}{2} \)
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Given the system of equations \[ \frac{1}{3}x^2 - \frac{1}{3}y^2 = 7 \] \[ 0.01x + 0.01y = 0.05 \] What is \( x - y \)?
Solution
Multiply all terms of the first equation by 3 to obtain
\[ x^2 - y^2 = 21 \]
Multiply all terms of the second equation by 100 to obtain
\[ x + y = 5 \]
Factor the left side of equation \( x^2 - y^2 = 21 \) and rewrite as
\[ (x + y)(x - y) = 21 \]
Substitute \( x + y \) by 5 in the above and solve for \( x - y \)
\[ 5(x - y) = 21 \implies x - y = \frac{21}{5} \]
Answer: \( \frac{21}{5} \)
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Function \( f \) is given by \( f(x) = x^2 + a x + b \), where \( a \) and \( b \) are real numbers. What are the values of \( a \) and \( b \) if the division of \( f(x) \) by \( x - 1 \) gives a remainder equal to -2 and the division of \( f(x) \) by \( x + 2 \) gives a remainder equal to -5?
Solution
Remainder theorem: The remainder of the division of a polynomial \( f(x) \) by \( x - a \) is \( f(a) \).
\[ f(1) = (1)^2 + a(1) + b = -2 \implies a + b = -3 \]
\[ f(-2) = (-2)^2 + a(-2) + b = -5 \implies -2a + b = -9 \]
Solve the system. Subtract the second equation from the first to obtain:
\[ 3a = 6 \implies a = 2 \]
Use \( a + b = -3 \) to find \( b \):
\[ 2 + b = -3 \implies b = -5 \]
Answer: \( a = 2 \) and \( b = -5 \)
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What are the values of the real numbers \( a, b \) and \( c \) if the equation \( -4x(x + 5) - 3(4x + 2) = a x^2 + b x + c \) is true for all values of \( x \)?
Solution
Expand the left side and group like terms.
\[ -4x^2 - 20x - 12x - 6 = a x^2 + b x + c \]
\[ -4x^2 - 32x - 6 = a x^2 + b x + c \]
Equate corresponding coefficients.
Answer: \( a = -4, b = -32, c = -6 \)
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What is the simplified form of the expression \( |x - 10| + |x - 12| \) for values of \( x \) such that \( 10 < x < 12 \)?
Solution
For \( 10 < x < 12 \), \( x - 10 > 0 \) and \( x - 12 < 0 \).
\[ |x - 10| = x - 10, \quad |x - 12| = -(x - 12) \]
\[ |x - 10| + |x - 12| = x - 10 - (x - 12) = 2 \]
Answer: \( 2 \) for \( 10 < x < 12 \)
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A function is defined by \( y = \frac{2x - 1}{x + 3} \). What is the value of \( x \) for \( y = -\frac{1}{4} \)?
Solution
Set \( y = -\frac{1}{4} \) and solve for \( x \).
\[ -\frac{1}{4} = \frac{2x - 1}{x + 3} \]
Cross multiply: \[ - (x + 3) = 4(2x - 1) \implies -x - 3 = 8x - 4 \implies -9x = -1 \]
Answer: \( x = \frac{1}{9} \)
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Which of the graphs below may be that of equation \( -3x + 3y = 3 \)?
Solution
Write in slope-intercept form:
\[ 3y = 3x + 3 \implies y = x + 1 \]
Slope = 1, y-intercept = 1 → corresponds to line L.
Answer: line L
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Which of the graphs below may be that of equation \( 2y - 2(x - 2)^2 - 2 = 0 \)?
Solution
Solve for \( y \):
\[ 2y = 2(x - 2)^2 + 2 \implies y = (x - 2)^2 + 1 \]
Parabola shifted 2 units right, 1 unit up → corresponds to graph M.
Answer: Parabola M
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What is the solution set for the equation \( |x - 3| = \sqrt{x + 17} \)?
Solution
Square both sides:
\[ (|x - 3|)^2 = (\sqrt{x + 17})^2 \implies (x - 3)^2 = x + 17 \]
\[ x^2 - 6x + 9 = x + 17 \implies x^2 - 7x - 8 = 0 \implies (x + 1)(x - 8) = 0 \]
Answer: \( \{-1, 8\} \)
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Find the solution set for \( x^2 = 7|x| - 10 \).
Solution
Rewrite: \( x^2 - 7|x| + 10 = 0 \). Let \( u = |x| \), \( u^2 = x^2 \):
\[ u^2 - 7u + 10 = 0 \implies (u - 5)(u - 2) = 0 \implies u = 5, u = 2 \]
\[ |x| = 5 \implies x = \pm 5, \quad |x| = 2 \implies x = \pm 2 \]
Answer: \( \{-5, -2, 2, 5\} \)
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Write the inequality \( \frac{3}{2} \le x \le \frac{5}{2} \) using one inequality symbol only.
Solution
Midpoint: \( \frac{1}{2}\left(\frac{3}{2} + \frac{5}{2}\right) = 2 \), distance = \( 0.5 \).
Answer: \( |x - 2| \le 0.5 \)
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What is the solution set for \( (x - 2)(x^2 - 7x + 13) - x + 2 = 0 \)?
Solution
Factor out \( x - 2 \):
\[ (x - 2)(x^2 - 7x + 13) - (x - 2) = 0 \implies (x - 2)(x^2 - 7x + 12) = 0 \]
\[ x - 2 = 0 \implies x = 2; \quad x^2 - 7x + 12 = 0 \implies (x - 3)(x - 4) = 0 \implies x = 3, 4 \]
Answer: \( \{2, 3, 4\} \)
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If \( f \) is a function, which of the functions below must have a graph symmetric with respect to the y-axis?
a) \( g(x) = (f(x))^2 \) b) \( h(x) = |f(x)| \) c) \( i(x) = f(x^2) \) d) \( j(x) = f(-x) \)
Solution
A function is even if \( f(-x) = f(x) \). Only \( i(x) = f(x^2) \) satisfies this for any \( f \):
\[ i(-x) = f((-x)^2) = f(x^2) = i(x) \]
Answer: c)
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Find all values of \( k \) for which \( -2|x - 4| - 2 = k + 1 \) has two solutions.
Solution
Isolate absolute value:
\[ |x - 4| = \frac{k + 3}{-2} \]
For two solutions, RHS must be positive: \[ \frac{k + 3}{-2} > 0 \implies k + 3 < 0 \implies k < -3 \]
Answer: \( k < -3 \)
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Find the value of \( x \) if \( (x + 2)^2 + 2(x + 2) + y = -2 \) and \( y - 2 = x \).
Solution
From \( y - 2 = x \), \( y = x + 2 \). Let \( u = x + 2 \), then \( y = u \).
\[ u^2 + 2u + u = -2 \implies u^2 + 3u + 2 = 0 \implies (u + 1)(u + 2) = 0 \implies u = -1, -2 \]
\[ x = u - 2 \implies x = -3 \text{ or } x = -4 \]
Answer: \( -3 \) and \( -4 \)
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Find \( r = \frac{f(x + h) - f(x)}{h} \) if \( f(x) = m x + b \).
Solution
\[ f(x + h) = m(x + h) + b \]
\[ r = \frac{m(x + h) + b - (mx + b)}{h} = \frac{mh}{h} = m \]
Answer: \( r = m \)
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What is the solution set for \( \frac{2}{w + 2} = \frac{4}{w+3} - \frac{1}{3} \)?
Solution
Multiply by LCM \( 3(w+2)(w+3) \):
\[ 6(w+3) = 12(w+2) - (w+2)(w+3) \]
\[ 6w + 18 = 12w + 24 - (w^2 + 5w + 6) \implies w^2 - w = 0 \implies w(w - 1) = 0 \]
Answer: \( \{0, 1\} \)
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The equations of the two parabolas are \( y = x^2 + A x + B \) and \( y = -x^2 + M x + N \). They are tangent (touch at one point) and \( A - M = 2 \). Find \( B - N \).
Solution
Set equations equal:
\[ x^2 + A x + B = -x^2 + M x + N \implies 2x^2 + (A - M)x + (B - N) = 0 \]
Tangency: discriminant \( D = 0 \): \[ (A - M)^2 - 8(B - N) = 0 \implies 4 - 8(B - N) = 0 \]
Answer: \( B - N = \frac{1}{2} \)
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If lines \( A x + B y = C \) and \( M x + N y = P \) are perpendicular and \( \frac{M}{N} = 5 \), find \( \frac{A}{B} \).
Solution
Slopes: \( m_1 = -\frac{A}{B} \), \( m_2 = -\frac{M}{N} = -5 \). Perpendicular: \( m_1 m_2 = -1 \).
\[ \left(-\frac{A}{B}\right)(-5) = -1 \implies 5\frac{A}{B} = -1 \implies \frac{A}{B} = -\frac{1}{5} \]
Answer: \( -\frac{1}{5} \)
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Solve for \( x \) in terms of \( K, L, M, N, P \): \( -\frac{K x - L}{M x - N} = P \).
Solution
\[ -Kx + L = P(Mx - N) \implies -Kx + L = PMx - PN \]
\[ L + PN = x(PM + K) \implies x = \frac{L + PN}{PM + K} \]
Answer: \( x = \frac{L + PN}{PM + K} \)
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The square root of a real number plus twice the same number is equal to 10. Find the number.
Solution
\[ \sqrt{x} + 2x = 10 \]
Let \( u = \sqrt{x} \), \( u^2 = x \): \( 2u^2 + u - 10 = 0 \implies (2u + 5)(u - 2) = 0 \implies u = 2 \) (since \( \sqrt{x} \ge 0 \)), so \( x = 4 \).
Answer: \( 4 \)
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The graph of \( f(x) = -x^2 + a \) and the line \( y = x - 2 \) intersect on the x-axis. Find \( a \).
Solution
x-intercept of line: \( 0 = x - 2 \implies x = 2 \). Point lies on parabola: \( 0 = -(2)^2 + a \implies a = 4 \).
Answer: \( a = 4 \)
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For what values of \( x \) is \( f(x) = \frac{x+2}{\sqrt{|x-2|-4}} \) not a real number?
Solution
Denominator undefined or zero: \( |x-2| - 4 \le 0 \implies |x-2| \le 4 \implies -2 \le x \le 6 \).
Answer: \( [-2, 6] \)
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What is \( 0.25x + 0.15y \) if \( 5x + 3y = 2 \)?
Solution
Multiply \( 5x + 3y = 2 \) by \( 0.05 \): \( 0.05(5x + 3y) = 0.1 \implies 0.25x + 0.15y = 0.1 \).
Answer: \( 0.1 \)
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If \( \frac{8 - 16i}{2 - 2i} \) is written as \( a + ib \), find \( b \).
Solution
Multiply numerator and denominator by conjugate \( 2 + 2i \):
\[ \frac{(8 - 16i)(2 + 2i)}{(2 - 2i)(2 + 2i)} = \frac{16 + 16i - 32i - 32i^2}{4 + 4} = \frac{48 - 16i}{8} = 6 - 2i \]
Answer: \( b = -2 \)
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Given \( 0.2(x + y)^2 = 4 \) and \( 0.5(x - y)^2 = 3 \), find \( xy \).
Solution
\[ (x + y)^2 = \frac{4}{0.2} = 20, \quad (x - y)^2 = \frac{3}{0.5} = 6 \]
Subtract: \( (x + y)^2 - (x - y)^2 = 20 - 6 \implies 4xy = 14 \implies xy = \frac{7}{2} \)
Answer: \( \frac{7}{2} \)
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There are 200 liters of water in a tank. It leaks at 0.25 L/min for one hour, then at 0.4 L/min. Find water left after \( t \) hours (\( t > 1 \)).
Solution
Leaked in first hour: \( 60 \times 0.25 = 15 \) L. After first hour: \( 60(t - 1) \times 0.4 \) L.
\[ q = 200 - 15 - 24(t - 1) \]
Answer: \( q = 185 - 24(t - 1) \) or \( q = 209 - 24t \)
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Zoe writes 30 pages. She writes 11 pages every 2h35m. How many hours to finish? Round to nearest hour.
Solution
Rate: \( \frac{11}{155} \) pages/min. Time: \( \frac{30}{11/155} = 30 \times \frac{155}{11} \approx 422.73 \) min ≈ 7.05 hours.
Answer: \( \approx 7 \) hours
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Car bought at \( \$50,000 \). Depreciates \( \$4000/year\) for 2 years, then \( \$6000/year \). Write price \( P(t) \) for \( t \ge 0 \) where \( t = 0 \) corresponds to 2 years after purchase.
Solution
Price after 2 years: \( 50000 - 2 \times 4000 = 42000 \). After that, \( P(t) = 42000 - 6000t \).
Answer: \( P(t) = 42000 - 6000t \)