Chapter 2 Study Guide and Review Geometry Answers Linear Relations and Functions

Endeavor It

2.1 The Rectangular Coordinate Systems and Graphs

1 .

$\mathrm{ten}$	$y=\frac{1}{2}x+2$	$\left(\mathrm{ten},y\right)$
$\mathrm{-2}$	$y=\frac{\mathrm{one}}{\mathrm{ii}}\left(\mathrm{-2}\right)+2=\mathrm{one}$	$\left(\mathrm{-2},\mathrm{ane}\right)$
$\mathrm{-i}$	$y=\frac{\mathrm{i}}{2}\left(\mathrm{-1}\right)+\mathrm{ii}=\frac{3}{2}$	$\left(-1,\frac{3}{2}\right)$
$0$	$y=\frac{1}{\mathrm{two}}\left(0\right)+\mathrm{two}=2$	$\left(0,2\right)$
$1$	$y=\frac{1}{2}\left(\mathrm{i}\right)+\mathrm{ii}=\frac{5}{2}$	$\left(1,\frac{5}{\mathrm{ii}}\right)$
$\mathrm{ii}$	$y=\frac{1}{\mathrm{ii}}\left(\mathrm{ii}\right)+2=3$	$\left(2,3\right)$

2 .

x-intercept is $\left(4,0\right);$ y-intercept is $\left(0,\mathrm{iii}\right).$

iv .

$\left(-\mathrm{five},\frac{5}{2}\right)$

2.two Linear Equations in One Variable

5 .

$x=-\frac{7}{17}.$ Excluded values are $x=-\frac{1}{2}$ and $x=-\frac{\mathrm{one}}{3}.$

x .

Horizontal line: $y=\mathrm{ii}$

xi .

Parallel lines: equations are written in slope-intercept form.

2.three Models and Applications

2 .

$C=2.5\mathrm{10}+\mathrm{iii},650$

4 .

$\mathrm{Fifty}=37$ cm, $\mathrm{West}=18$ cm

2.iv Complex Numbers

1 .

$\sqrt{\mathrm{-24}}=0+2i\sqrt{6}$

3 .

$(\mathrm{three}\mathrm{-4}i)-(2+\mathrm{v}i)=1\mathrm{-9}i$

2.5 Quadratic Equations

1 .

$\left(x-6\right)\left(\mathrm{10}+1\right)=0;x=\mathrm{half-dozen},x=-1$

2 .

$\left(x\mathrm{-7}\right)\left(\mathrm{ten}+\mathrm{iii}\right)=0,$ $x=\mathrm{seven},$ $x=\mathrm{-3.}$

three .

$\left(x+5\right)\left(x\mathrm{-5}\right)=0,$ $\mathrm{10}=\mathrm{-5},$ $x=\mathrm{five.}$

iv .

$\left(3\mathrm{10}+2\right)\left(\mathrm{four}\mathrm{ten}+\mathrm{i}\right)=0,$ $x=-\frac{2}{\mathrm{three}},$ $x=-\frac{1}{4}$

five .

$x=0,x=\mathrm{-10},x=\mathrm{-1}$

eight .

$x=-\frac{2}{3},$ $x=\frac{1}{3}$

2.6 Other Types of Equations

four .

$0,$ $\frac{\mathrm{one}}{2},$ $-\frac{1}{2}$

5 .

$1;$ extraneous solution $-\frac{2}{\mathrm{nine}}$

6 .

$\mathrm{-2};$ extraneous solution $\mathrm{-1}$

10 .

$\mathrm{-1},$ $0$ is non a solution.

2.7 Linear Inequalities and Absolute Value Inequalities

2 .

$\left(-\infty ,\mathrm{-2}\right)\cup \left[3,\infty \right)$

six .

$\left[-\frac{\mathrm{three}}{14},\infty \right)$

7 .

$\mathrm{six}<x\le 9\text{}\text{or}\left(6,9\right]$

8 .

$\left(-\frac{\mathrm{one}}{8},\frac{\mathrm{i}}{2}\right)$

x .

$k\le 1$ or $k\ge \mathrm{seven};$ in interval annotation, this would be $(-\infty ,1]\cup [7,\infty ).$

2.1 Section Exercises

1 .

Answers may vary. Yes. It is possible for a point to exist on the x-axis or on the y-centrality and therefore is considered to NOT be in one of the quadrants.

3 .

The y-intercept is the point where the graph crosses the y-axis.

5 .

The 10-intercept is $\left(\mathrm{ii},0\right)$ and the y-intercept is $\left(0,\mathrm{vi}\right).$

7 .

The ten-intercept is $\left(\mathrm{ii},0\right)$ and the y-intercept is $\left(0,\mathrm{-iii}\right).$

nine .

The x-intercept is $\left(3,0\right)$ and the y-intercept is $\left(0,\frac{\mathrm{nine}}{\mathrm{viii}}\right).$

23 .

$\left(3,\frac{-3}{2}\right)$

31 .

not collinear

33 .

$\text{A:}\left(\mathrm{-3},2\right),\text{B:}\left(1,3\right),\text{C:}\left(\mathrm{four},0\right)$

35 .

$x$	$y$
$\mathrm{-3}$	1
0	2
iii	iii
6	4

49 .

$x=0y=\mathrm{-two}$

53 .

$x=-1.667y=0$

55 .

$\mathrm{fifteen}-\mathrm{eleven.2}=3.8\text{mi}$ shorter

59 .

Midpoint of each diagonal is the same indicate $(2,\mathrm{–2})$ . Annotation this is a feature of rectangles, but not other quadrilaterals.

2.2 Section Exercises

one .

It means they have the same slope.

3 .

The exponent of the $x$ variable is ane. It is chosen a first-caste equation.

5 .

If we insert either value into the equation, they make an expression in the equation undefined (goose egg in the denominator).

17 .

$\mathrm{10}\ne \mathrm{-4};$ $x=\mathrm{-3}$

19 .

$x\ne 1;$ when we solve this we get $x=1,$ which is excluded, therefore NO solution

21 .

$\mathrm{10}\ne 0;$ $x=-\frac{5}{\mathrm{ii}}$

23 .

$y=-\frac{\mathrm{four}}{\mathrm{v}}\mathrm{10}+\frac{\mathrm{xiv}}{5}$

25 .

$y=-\frac{3}{4}x+\mathrm{two}$

27 .

$y=\frac{1}{2}\mathrm{ten}+\frac{\mathrm{v}}{2}$

37 .

Parallel

39 .

Perpendicular

45 .

${\mathrm{one\; thousand}}_{\mathrm{one}}=-\frac{1}{3},m_2=3;\text{Perpendicular}\text{.}$

47 .

$y=0.245x-\mathrm{45.662.}$ Answers may vary. $y_{\text{min}}=\mathrm{-50},y_{\text{max}}=\mathrm{-40}$

49 .

$y=-\mathrm{two.333}x+\mathrm{half-dozen.667.}$ Answers may vary. $y_{\min }=\mathrm{-10},y_{\max }=10$

51 .

$y=-\frac{A}{B}x+\frac{C}{B}$

53 .

$\begin{array}{l}\text{The slope for}(\mathrm{-1},\mathrm{ane})\text{to}(0,\mathrm{iv})\text{is}3.\\ \text{The slope for}(\mathrm{-one},1)\text{to}(2,0)\text{is}\frac{-\mathrm{ane}}{\mathrm{iii}}.\\ \text{The slope for}(2,0)\text{to}(\mathrm{three},3)\text{is}3.\\ \text{The slope for}(0,4)\text{to}(\mathrm{three},3)\text{is}\frac{-1}{\mathrm{three}}.\end{array}$

Yes they are perpendicular.

2.3 Department Exercises

1 .

Answers may vary. Possible answers: We should ascertain in words what our variable is representing. Nosotros should declare the variable. A heading.

vii .

Ann: $23;$ Beth: $46$

21 .

She traveled for 2 h at twenty mi/h, or xl miles.

23 .

$5,000 at 8% and $15,000 at 12%

25 .

$B=100+.05x$

31 .

$r=\frac{4}{\mathrm{five}}$ or 0.viii

33 .

$W=\frac{P-2L}{2}=\frac{58-\mathrm{ii}(15)}{2}=14$

35 .

$f=\frac{pq}{p+q}=\frac{\mathrm{eight}(13)}{\mathrm{eight}+13}=\frac{104}{21}$

39 .

$h=\frac{2A}{b_1+b_{\mathrm{two}}}$

41 .

length = 360 ft; width = 160 ft

45 .

$A=88\text{in}{.}^2$

49 .

$h=\frac{V}{\pi r^{\mathrm{two}}}$

2.iv Section Exercises

1 .

Add the real parts together and the imaginary parts together.

3 .

Possible respond: $i$ times $i$ equals -1, which is not imaginary.

9 .

$-\frac{23}{29}+\frac{15}{29}i$

33 .

$\frac{\mathrm{two}}{\mathrm{five}}+\frac{11}{5}i$

45 .

${\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}^6=\mathrm{-one}$

55 .

$\frac{\mathrm{nine}}{2}-\frac{9}{2}i$

two.5 Section Exercises

one .

It is a 2d-degree equation (the highest variable exponent is two).

3 .

Nosotros want to take advantage of the zero property of multiplication in the fact that if $a\cdot b=0$ then it must follow that each factor separately offers a solution to the product being nil: $a=0or\text{b}=0.$

five .

One, when no linear term is present (no ten term), such equally ${\mathrm{10}}^2=16.$ Two, when the equation is already in the form ${(ax+b)}^2=d.$

9 .

$x=\frac{-5}{2},$ $x=\frac{-1}{3}$

thirteen .

$x=\frac{-\mathrm{three}}{2},$ $\mathrm{10}=\frac{3}{\mathrm{two}}$

17 .

$x=0,$ $x=\frac{-3}{\mathrm{seven}}$

25 .

$\mathrm{ten}=\mathrm{-ii},$ $x=\mathrm{xi}$

29 .

$z=\frac{2}{3},$ $z=-\frac{\mathrm{one}}{\mathrm{two}}$

31 .

$\mathrm{10}=\frac{\mathrm{three}\pm \sqrt{17}}{4}$

39 .

$x=\frac{-1\pm \sqrt{17}}{2}$

41 .

$\mathrm{ten}=\frac{5\pm \sqrt{13}}{\mathrm{six}}$

43 .

$x=\frac{-1\pm \sqrt{17}}{8}$

45 .

$x\approx 0.131$ and $x\approx \mathrm{two.535}$

47 .

$x\approx -\mathrm{6.seven}$ and $x\approx \mathrm{1.seven}$

49 .

$\begin{array}{ccc}\hfill a{\mathrm{10}}^2+bx+c& =& 0\hfill \\ \hfill {\mathrm{ten}}^2+\frac{b}{a}\mathrm{10}& =& \frac{-c}{a}\hfill \\ \hfill x^2+\frac{b}{a}x+\frac{b^2}{4a^2}& =& \frac{-c}{a}+\frac{b}{4a^2}\hfill \\ \hfill {\left(x+\frac{b}{2a}\right)}^2& =& \frac{b^2-4ac}{4a^2}\hfill \\ \hfill \mathrm{10}+\frac{b}{2a}& =& \pm \sqrt{\frac{b^{\mathrm{two}}-4ac}{\mathrm{iv}{a}^{\mathrm{two}}}}\hfill \\ \hfill x& =& \frac{-b\pm \sqrt{b^2-4ac}}{\mathrm{ii}a}\hfill \end{array}$

51 .

$x(x+10)=119;$ vii ft. and 17 ft.

53 .

maximum at $\mathrm{10}=\mathrm{lxx}$

55 .

The quadratic equation would be $(100\mathrm{ten}\mathrm{-0.five}{x}^{\mathrm{two}})-(60x+300)=300.$ The ii values of $\mathrm{10}$ are twenty and lx.

2.6 Section Exercises

1 .

This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it non to be a solution in the original equation.

iii .

He or she is probably trying to enter negative ix, but taking the square root of $\mathrm{-ix}$ is not a real number. The negative sign is in forepart of this, and so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in $\mathrm{-27.}$

5 .

A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.

xi .

$x=\mathrm{viii},x=27$

fifteen .

$y=0,\frac{3}{2},\frac{-3}{\mathrm{ii}}$

19 .

$x=\frac{2}{\mathrm{five}},\mathrm{\pm iii}i$

31 .

$x=\frac{-\mathrm{five}}{4},\frac{7}{\mathrm{iv}}$

37 .

$x=\mathrm{i},\mathrm{-1},\mathrm{three},-3$

45 .

$\mathrm{10}=4,\mathrm{six},\mathrm{-6},\mathrm{-viii}$

2.7 Section Exercises

1 .

When we divide both sides past a negative it changes the sign of both sides and so the sense of the inequality sign changes.

v .

We showtime by finding the x-intercept, or where the function = 0. Once we take that bespeak, which is $(\mathrm{iii},0),$ we graph to the right the straight line graph $y=x\mathrm{-iii},$ and then when we draw it to the left nosotros plot positive y values, taking the absolute value of them.

7 .

$\left(-\infty ,\frac{3}{4}\right]$

9 .

$\left[-\frac{13}{2},\infty \right)$

13 .

$\left(-\infty ,-\frac{37}{\mathrm{iii}}\right]$

15 .

All real numbers $\left(-\infty ,\infty \right)$

17 .

$\left(-\infty ,-\frac{10}{3}\right)\cup \left(4,\infty \right)$

nineteen .

$\left(-\infty ,\mathrm{-iv}\right]\cup \left[\mathrm{eight},+\infty \right)$

27 .

$\left[\mathrm{-10},12\right]$

29 .

$\begin{array}{ll}\mathrm{10}>-\mathrm{six}\text{and}x>-2\hfill & \text{Accept the intersection of ii sets}.\hfill \\ \mathrm{ten}>-\mathrm{ii},(-2,+\infty )\hfill & \hfill \end{array}$

31 .

$\begin{array}{ll}\mathrm{ten}<-\mathrm{iii}\mathrm{or}x\ge \mathrm{ane}\hfill & \text{Take the wedlock of the two sets}.\hfill \\ (-\infty ,-3){{{\displaystyle \cup }}^{\text{}}}^{\text{}}[1,\infty )\hfill & \hfill \end{array}$

33 .

$\left(-\infty ,\mathrm{-i}\right)\cup \left(\mathrm{iii},\infty \right)$

35 .

$\left[\mathrm{-11},\mathrm{-3}\right]$

37 .

It is never less than zero. No solution.

39 .

Where the blue line is above the orange line; point of intersection is $x=-3.$

$\left(-\infty ,\mathrm{-three}\right)$

41 .

Where the blue line is above the orangish line; ever. All existent numbers.

$(-\infty ,-\infty )$

47 .

$\{x|\mathrm{10}<\mathrm{half\; dozen}\}$

49 .

$\{x|\mathrm{-three}\le x<\mathrm{five}\}$

55 .

Where the blue is below the orange; e'er. All real numbers. $(-\infty ,+\infty ).$

57 .

Where the bluish is beneath the orange; $\left(1,7\right).$

63 .

$\begin{array}{l}80\le T\le 120\\ 1,600\le 20T\le \mathrm{two},400\end{array}$

$\left[1,600,2,400\right]$

Review Exercises

1 .

ten-intercept: $\left(3,0\right);$ y-intercept: $\left(0,\mathrm{-four}\right)$

three .

$y=\frac{5}{\mathrm{three}}\mathrm{ten}+4$

9 .

midpoint is $\left(2,\frac{23}{\mathrm{ii}}\right)$

19 .

$y=\frac{1}{\mathrm{six}}\mathrm{ten}+\frac{4}{\mathrm{iii}}$

21 .

$y=\frac{2}{3}x+6$

23 .

females 17, males 56

27 .

$\mathrm{10}=-\frac{\mathrm{three}}{4}\pm \frac{i\sqrt{47}}{\mathrm{four}}$

29 .

horizontal component $\mathrm{-2};$ vertical component $\mathrm{-i}$

47 .

$x=\frac{-1\pm \sqrt{5}}{\mathrm{four}}$

49 .

$\mathrm{10}=\frac{2}{5},\frac{-\mathrm{one}}{3}$

59 .

$x=\frac{11}{2},\frac{\mathrm{-17}}{\mathrm{ii}}$

63 .

$\left[\frac{-10}{3},\mathrm{two}\right]$

67 .

$\left(-\frac{4}{3},\frac{\mathrm{ane}}{5}\right)$

69 .

Where the blue is beneath the orange line; point of intersection is $x=\mathrm{iii.5.}$

$\left(\mathrm{three.5},\infty \right)$

Practice Test

1 .

$y=\frac{3}{\mathrm{two}}\mathrm{10}+2$

iii .

$\left(0,\mathrm{-3}\right)$ $\left(\mathrm{iv},0\right)$

9 .

$x\ne \mathrm{-four},2;$ $x=\frac{-\mathrm{five}}{\mathrm{ii}},1$

15 .

$y=\frac{\mathrm{-5}}{9}x-\frac{\mathrm{ii}}{9}$

17 .

$y=\frac{5}{2}x-\mathrm{four}$

21 .

$\frac{\mathrm{v}}{\mathrm{xiii}}-\frac{\mathrm{fourteen}}{13}i$

25 .

$x=\frac{1}{2}\pm \frac{\sqrt{2}}{2}$

29 .

$x=\frac{\mathrm{i}}{2},2,\mathrm{-2}$

tysonfrotingle.blogspot.com

Source: https://openstax.org/books/college-algebra/pages/chapter-2

Share this post