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Questions and Answers

Solve $\frac{2}{5}y = \frac{3}{14}$

y = \frac{28}{15}

Name the property illustrated by $7(9+1)=(9+1)7$

Commutative property of addition

Solve $18 = 3 |4x-10|$

x = 4, 1

Write a linear equation.

<p>y = 3x + 7</p> Signup and view all the answers

Write $-3y = -1 + 5x$ in standard form.

<p>5x + 3y = 1</p> Signup and view all the answers

Find the x-intercept of $4x-2y=8$.

<p>x = 2</p> Signup and view all the answers

What is the slope of the line $x = -2$?

<p>no slope</p> Signup and view all the answers

What is the slope of the line $y = -2$?

<p>0</p> Signup and view all the answers

A system of linear equations has how many solutions?

<p>None, one or infinitely many</p> Signup and view all the answers

The system of equations $y = -3x + 5$ and $y = 3x - 7$ has how many solutions? What are they?

<p>One, (2, -1)</p> Signup and view all the answers

Find the minimum and maximum value of $f(x,y) = 3x + y$ for the feasible region above.

<p>min = 0, max = 6</p> Signup and view all the answers

Determine whether $f(x) = 4x^2 - 16x + 6$ has a maximum or a minimum value and find that value.

<p>(2, -10) min</p> Signup and view all the answers

Write the quadratic equation that has the roots -2 and $\frac{1}{5}$.

<p>5x^2 + 9x - 2 = 0</p> Signup and view all the answers

Solve by using the quadratic formula $3x^2 = 5x - 1$.

<p>$\frac{5 + \sqrt{13}}{6}$</p> Signup and view all the answers

Simplify $(4 - 12i) - (-8 + 4i)$.

<p>12 - 16i</p> Signup and view all the answers

Find the points for #39.

<p>(2,4) other points -&gt; (0, -4)</p> Signup and view all the answers

Find the points for #40.

<p>(-1, -4) other points -&gt; (0, -3)</p> Signup and view all the answers

Simplify $(5x-4)^2$.

<p>25x^2 - 40x + 16</p> Signup and view all the answers

Use synthetic division to simplify $(3x^3 - 2x + 5) / (x - 2)$.

<p>3x^2 + 6x + 10 + \frac{25}{x-2}</p> Signup and view all the answers

How many real zeros does the graph of the function have?

<p>3</p> Signup and view all the answers

How many real zeros are located on the graph?

<p>1, 3, between 4 &amp; 5</p> Signup and view all the answers

Find $(f-g)(x)$ for $f(x) = x^2 + 8x$ and $g(x) = 3x + 5$.

<p>x^2 + 5x - 5</p> Signup and view all the answers

Find the inverse of $f(x) = 3 + 5x$.

<p>$\frac{x-3}{5}$</p> Signup and view all the answers

Graph $y > \sqrt{x} + 3$.

<p>(0, 3) shade above</p> Signup and view all the answers

Simplify $\sqrt[3]{256t^4}$.

<p>4t\sqrt{4t}</p> Signup and view all the answers

Simplify $\sqrt{32} - \sqrt{18} + \sqrt{54} + \sqrt{150}$.

<p>$\sqrt{2} + 8\sqrt{16}$</p> Signup and view all the answers

Simplify $\frac{5}{2 - \sqrt{3}}$.

<p>$10 + 5\sqrt{3}$</p> Signup and view all the answers

Simplify $\sqrt{5} + \sqrt{20} - \sqrt{27} + \sqrt{147}$.

<p>$3\sqrt{5} + 4\sqrt{3}$</p> Signup and view all the answers

Find the domain and range of the function $y = (\frac{1}{2})(2)^x$.

<p>D: all real #'s, R: y &gt; 0</p> Signup and view all the answers

Write an exponential function that represents growth.

<p>$y = \frac{1}{2}(\frac{5}{3})^x$ —&gt; $y = \frac{1}{20}(\frac{5}{2})^x$</p> Signup and view all the answers

Solve $32^{x+3} = 4^{2x+7}$.

<p>x = -1</p> Signup and view all the answers

Solve $64^{x} < 32^{x+2}$.

<p>x &lt; 10</p> Signup and view all the answers

Solve $\log_{\frac{1}{8}} x = -1$.

<p>x = 8</p> Signup and view all the answers

Where is $\frac{x^2 - 4x + 4}{2x^2 - 3x - 2}$ undefined?

<p>x /= (-\frac{1}{2}), 2</p> Signup and view all the answers

Simplify $\frac{(m + 2t - 3)}{(t^2 - 1)} * \frac{(3t - 3)}{(t^2 - 4t + 3)}$.

<p>$\frac{5}{3(m - 2t)}$</p> Signup and view all the answers

Simplify $\frac{3b^2 - 12}{(6b^2 + 12b)} / \frac{(5b - 10)}{(10b^2 + 20b)}$.

<p>b + 2</p> Signup and view all the answers

Simplify $\frac{30}{(m^2 - 25)} + \frac{3}{(m-5)}$.

<p>$\frac{3m + 45}{(m + 5)(m - 5)}$</p> Signup and view all the answers

Find the LCM of $7m - 21$ and $14m - 42$.

<p>14(m - 3)</p> Signup and view all the answers

Find the LCM of $t^2 - t - 12$ and $14m - 42$.

<p>(t - 4)(t + 3)(t + 6)</p> Signup and view all the answers

If y varies inversely as x and y = 5 when x = 5 find y when x = 45.

<p>$\frac{5}{9}$</p> Signup and view all the answers

Solve $7 - \frac{3}{m} > \frac{18}{m}$.

<p>m &gt; 3</p> Signup and view all the answers

Write the equation of the parabola in standard form $y = 2x^2 - 8x + 1$.

<p>$y = 2(x - 2)^2 - 7$</p> Signup and view all the answers

Write the equation for a circle if the endpoints of the diameter are (-7, 1) and (5, 1).

<p>$(x + 1)^2 + (y - 1)^2$</p> Signup and view all the answers

Find the 20th term of the arithmetic sequence in which $a_1 = 5$ and d = 4.

<p>81</p> Signup and view all the answers

Write an equation for the nth term of the arithmetic sequence -7, -2, 3, 8,...

<p>$a_n = 5n - 12$</p> Signup and view all the answers

Find two arithmetic means for 6, __, __, 30.

<p>14, 22</p> Signup and view all the answers

Find $S_n$ for the arithmetic series in which $a_1 = 3$, d = $\frac{1}{2}$, and $a_n = \frac{17}{2}$.

<p>69</p> Signup and view all the answers

Find $^22E(50-2x)$ when x = 18.

<p>50</p> Signup and view all the answers

Find the sixth term of the geometric sequence for which $a_1 = 4$ and r = 3.

<p>972</p> Signup and view all the answers

Find four geometric means for 486, __, __, __, __, 2.

<p>164, 54, 18, 6</p> Signup and view all the answers

Write 0.63 as a fraction.

<p>7/11</p> Signup and view all the answers

Write 0.735 as a fraction.

<p>245/333</p> Signup and view all the answers

Use binomial theorem to find the third term in $(x + 3y)^5$.

<p>90x^3y^2</p> Signup and view all the answers

A binomial distribution has a 65% rate of success in 15 trials. What is the probability to get exactly 12 successes?

<p>0.11096</p> Signup and view all the answers

State if a binomial distribution exists. If so, give a random variable, n, p, and q.

<p>Yes, only two answers: n=15, p=0.75, q=0.25</p> Signup and view all the answers

Find the range of variables that represents the middle 95% of the distribution.

<p>11.2 &lt; x &lt; 21.6</p> Signup and view all the answers

What percent of the data will be less than 19?

<p>84%</p> Signup and view all the answers

Solve $x = \tan^{-1}(-\sqrt{3})$.

<p>300°, 120°</p> Signup and view all the answers

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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