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Questions and Answers
Solve $\frac{2}{5}y = \frac{3}{14}$
Solve $\frac{2}{5}y = \frac{3}{14}$
y = \frac{28}{15}
Name the property illustrated by $7(9+1)=(9+1)7$
Name the property illustrated by $7(9+1)=(9+1)7$
Commutative property of addition
Solve $18 = 3 |4x-10|$
Solve $18 = 3 |4x-10|$
x = 4, 1
Write a linear equation.
Write a linear equation.
Write $-3y = -1 + 5x$ in standard form.
Write $-3y = -1 + 5x$ in standard form.
Find the x-intercept of $4x-2y=8$.
Find the x-intercept of $4x-2y=8$.
What is the slope of the line $x = -2$?
What is the slope of the line $x = -2$?
What is the slope of the line $y = -2$?
What is the slope of the line $y = -2$?
A system of linear equations has how many solutions?
A system of linear equations has how many solutions?
The system of equations $y = -3x + 5$ and $y = 3x - 7$ has how many solutions? What are they?
The system of equations $y = -3x + 5$ and $y = 3x - 7$ has how many solutions? What are they?
Find the minimum and maximum value of $f(x,y) = 3x + y$ for the feasible region above.
Find the minimum and maximum value of $f(x,y) = 3x + y$ for the feasible region above.
Determine whether $f(x) = 4x^2 - 16x + 6$ has a maximum or a minimum value and find that value.
Determine whether $f(x) = 4x^2 - 16x + 6$ has a maximum or a minimum value and find that value.
Write the quadratic equation that has the roots -2 and $\frac{1}{5}$.
Write the quadratic equation that has the roots -2 and $\frac{1}{5}$.
Solve by using the quadratic formula $3x^2 = 5x - 1$.
Solve by using the quadratic formula $3x^2 = 5x - 1$.
Simplify $(4 - 12i) - (-8 + 4i)$.
Simplify $(4 - 12i) - (-8 + 4i)$.
Find the points for #39.
Find the points for #39.
Find the points for #40.
Find the points for #40.
Simplify $(5x-4)^2$.
Simplify $(5x-4)^2$.
Use synthetic division to simplify $(3x^3 - 2x + 5) / (x - 2)$.
Use synthetic division to simplify $(3x^3 - 2x + 5) / (x - 2)$.
How many real zeros does the graph of the function have?
How many real zeros does the graph of the function have?
How many real zeros are located on the graph?
How many real zeros are located on the graph?
Find $(f-g)(x)$ for $f(x) = x^2 + 8x$ and $g(x) = 3x + 5$.
Find $(f-g)(x)$ for $f(x) = x^2 + 8x$ and $g(x) = 3x + 5$.
Find the inverse of $f(x) = 3 + 5x$.
Find the inverse of $f(x) = 3 + 5x$.
Graph $y > \sqrt{x} + 3$.
Graph $y > \sqrt{x} + 3$.
Simplify $\sqrt[3]{256t^4}$.
Simplify $\sqrt[3]{256t^4}$.
Simplify $\sqrt{32} - \sqrt{18} + \sqrt{54} + \sqrt{150}$.
Simplify $\sqrt{32} - \sqrt{18} + \sqrt{54} + \sqrt{150}$.
Simplify $\frac{5}{2 - \sqrt{3}}$.
Simplify $\frac{5}{2 - \sqrt{3}}$.
Simplify $\sqrt{5} + \sqrt{20} - \sqrt{27} + \sqrt{147}$.
Simplify $\sqrt{5} + \sqrt{20} - \sqrt{27} + \sqrt{147}$.
Find the domain and range of the function $y = (\frac{1}{2})(2)^x$.
Find the domain and range of the function $y = (\frac{1}{2})(2)^x$.
Write an exponential function that represents growth.
Write an exponential function that represents growth.
Solve $32^{x+3} = 4^{2x+7}$.
Solve $32^{x+3} = 4^{2x+7}$.
Solve $64^{x} < 32^{x+2}$.
Solve $64^{x} < 32^{x+2}$.
Solve $\log_{\frac{1}{8}} x = -1$.
Solve $\log_{\frac{1}{8}} x = -1$.
Where is $\frac{x^2 - 4x + 4}{2x^2 - 3x - 2}$ undefined?
Where is $\frac{x^2 - 4x + 4}{2x^2 - 3x - 2}$ undefined?
Simplify $\frac{(m + 2t - 3)}{(t^2 - 1)} * \frac{(3t - 3)}{(t^2 - 4t + 3)}$.
Simplify $\frac{(m + 2t - 3)}{(t^2 - 1)} * \frac{(3t - 3)}{(t^2 - 4t + 3)}$.
Simplify $\frac{3b^2 - 12}{(6b^2 + 12b)} / \frac{(5b - 10)}{(10b^2 + 20b)}$.
Simplify $\frac{3b^2 - 12}{(6b^2 + 12b)} / \frac{(5b - 10)}{(10b^2 + 20b)}$.
Simplify $\frac{30}{(m^2 - 25)} + \frac{3}{(m-5)}$.
Simplify $\frac{30}{(m^2 - 25)} + \frac{3}{(m-5)}$.
Find the LCM of $7m - 21$ and $14m - 42$.
Find the LCM of $7m - 21$ and $14m - 42$.
Find the LCM of $t^2 - t - 12$ and $14m - 42$.
Find the LCM of $t^2 - t - 12$ and $14m - 42$.
If y varies inversely as x and y = 5 when x = 5 find y when x = 45.
If y varies inversely as x and y = 5 when x = 5 find y when x = 45.
Solve $7 - \frac{3}{m} > \frac{18}{m}$.
Solve $7 - \frac{3}{m} > \frac{18}{m}$.
Write the equation of the parabola in standard form $y = 2x^2 - 8x + 1$.
Write the equation of the parabola in standard form $y = 2x^2 - 8x + 1$.
Write the equation for a circle if the endpoints of the diameter are (-7, 1) and (5, 1).
Write the equation for a circle if the endpoints of the diameter are (-7, 1) and (5, 1).
Find the 20th term of the arithmetic sequence in which $a_1 = 5$ and d = 4.
Find the 20th term of the arithmetic sequence in which $a_1 = 5$ and d = 4.
Write an equation for the nth term of the arithmetic sequence -7, -2, 3, 8,...
Write an equation for the nth term of the arithmetic sequence -7, -2, 3, 8,...
Find two arithmetic means for 6, __, __, 30.
Find two arithmetic means for 6, __, __, 30.
Find $S_n$ for the arithmetic series in which $a_1 = 3$, d = $\frac{1}{2}$, and $a_n = \frac{17}{2}$.
Find $S_n$ for the arithmetic series in which $a_1 = 3$, d = $\frac{1}{2}$, and $a_n = \frac{17}{2}$.
Find $^22E(50-2x)$ when x = 18.
Find $^22E(50-2x)$ when x = 18.
Find the sixth term of the geometric sequence for which $a_1 = 4$ and r = 3.
Find the sixth term of the geometric sequence for which $a_1 = 4$ and r = 3.
Find four geometric means for 486, __, __, __, __, 2.
Find four geometric means for 486, __, __, __, __, 2.
Write 0.63 as a fraction.
Write 0.63 as a fraction.
Write 0.735 as a fraction.
Write 0.735 as a fraction.
Use binomial theorem to find the third term in $(x + 3y)^5$.
Use binomial theorem to find the third term in $(x + 3y)^5$.
A binomial distribution has a 65% rate of success in 15 trials. What is the probability to get exactly 12 successes?
A binomial distribution has a 65% rate of success in 15 trials. What is the probability to get exactly 12 successes?
State if a binomial distribution exists. If so, give a random variable, n, p, and q.
State if a binomial distribution exists. If so, give a random variable, n, p, and q.
Find the range of variables that represents the middle 95% of the distribution.
Find the range of variables that represents the middle 95% of the distribution.
What percent of the data will be less than 19?
What percent of the data will be less than 19?
Solve $x = \tan^{-1}(-\sqrt{3})$.
Solve $x = \tan^{-1}(-\sqrt{3})$.
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Study Notes
Algebraic Equations and Solutions
- Solving the equation ( \frac{2}{5}y = \frac{3}{14} ) yields ( y = \frac{28}{15} ).
- The system of equations ( y = -3x + 5 ) and ( y = 3x - 7 ) has one solution, which is the point ( (2, -1) ).
- For ( 18 = 3 |4x-10| ), the solutions are ( x = 4 ) and ( x = 1 ).
Linear and Quadratic Functions
- A linear equation example is ( y = 3x + 7 ).
- The standard form transformation of ( -3y = -1 + 5x ) results in ( 5x + 3y = 1 ).
- The quadratic function ( f(x) = 4x^2 - 16x + 6 ) has a minimum value at the vertex ( (2, -10) ).
Slope and Intercepts
- The line represented by ( x = -2 ) is vertical and has no slope.
- The line ( y = -2 ) is horizontal with a slope of ( 0 ).
- The x-intercept of ( 4x - 2y = 8 ) is found at ( x = 2 ).
Inequalities and Graphing
- To graph ( y > \sqrt{x} + 3 ), start at point ( (0, 3) ) and shade above the curve.
- The domain of ( y = \frac{1}{2}(2)^x ) is all real numbers, and the range is ( y > 0 ).
Exponential and Logarithmic Functions
- An exponential growth function can be represented as ( y = \frac{1}{2} \left(\frac{5}{3}\right)^x ).
- Solving ( \log_{\frac{1}{8}} x = -1 ) yields ( x = 8 ).
Simplification Techniques
- Simplifying ( (4 - 12i) - (-8 + 4i) ) results in ( 12 - 16i ).
- The expression ( \sqrt{32} - \sqrt{18} + \sqrt{54} + \sqrt{150} ) simplifies to ( \sqrt{2} + 8\sqrt{16} ).
Sequences and Series
- The 20th term of an arithmetic sequence starting at ( 5 ) with a common difference of ( 4 ) is ( 81 ).
- The formula for the nth term of the arithmetic sequence ( -7, -2, 3, 8 ) is ( a_n = 5n - 12 ).
Circle and Parabola Equations
- The standard equation for a circle with endpoints of the diameter at ( (-7, 1) ) and ( (5, 1) ) is ( (x + 1)^2 + (y - 1)^2 = r^2 ) where ( r ) is the radius.
- The equation of the parabola ( y = 2x^2 - 8x + 1 ) in standard form is ( y = 2(x - 2)^2 - 7 ).
Binomial Theorem and Distributions
- Using the binomial theorem, the third term in ( (x + 3y)^5 ) is ( 90x^3y^2 ).
- A binomial distribution example includes a success rate of ( 65% ) in ( 15 ) trials, giving a probability of exactly ( 12 ) successes as ( 0.11096 ).
Miscellaneous
- Write ( 0.63 ) as a fraction: ( \frac{7}{11} ).
- Finding the sixth term for a geometric sequence where ( a_1 = 4 ) and ( r = 3 ) results in ( 972 ).
- The range representing the middle ( 95% ) of a distribution is ( 11.2 < x < 21.6 ).
- The value of ( m ) in the inequality ( 7 - \frac{3}{m} > \frac{18}{m} ) is ( m > 3 ).
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