6.2 Graphing linear inequalities (Page 3/3)

<< Chapter < Page

Algebra i for the community

Page 3 / 3

Chapter >> Page >

Practice set a

Graph $3 x + y = 3$ using the intercept method.

An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

When $x = 0$ , $y = 3$ ; when $y = 0$ , $x = 1$

A graph of a line passing through two points with coordinates zero, three and one, zero.

Graphing using any two or more points

The graphs we have constructed so far have been done by finding two particular points, the intercepts. Actually, any two points will do. We chose to use the intercepts because they are usually the easiest to work with. In the next example, we will graph two equations using points other than the intercepts. We’ll use three points, the extra point serving as a check.

Sample set b

$x - 3 y = - 10$ .
We can find three points by choosing three $x -values$ and computing to find the corresponding $y -values$ . We’ll put our results in a table for ease of reading.

Since we are going to choose $x -values$ and then compute to find the corresponding $y -values$ , it will be to our advantage to solve the given equation for $y$ .

$\begin{array}{r} x - 3 y & = & - 10 & Subtract x from both sides . \\ - 3 y & = & - x - 10 & Divide both sides by - 3. \\ y & = & \frac{1}{3} x + \frac{10}{3} \end{array}$

$x$	$y$	$(x, y)$
1	If $x = 1$ , then $y = \frac{1}{3} (1) + \frac{10}{3} = \frac{1}{3} + \frac{10}{3} = \frac{11}{3}$	$(1, \frac{11}{3})$
$- 3$	If $x = - 3$ , then $y = \frac{1}{3} (- 3) + \frac{10}{3} = - 1 + \frac{10}{3} = \frac{7}{3}$	$(- 3, \frac{7}{3})$
3	If $x = 3$ , then $y = \frac{1}{3} (3) + \frac{10}{3} = 1 + \frac{10}{3} = \frac{13}{3}$	$(3, \frac{13}{3})$

Thus, we have the three ordered pairs (points), $(1, \frac{11}{3})$ , $(- 3, \frac{7}{3})$ , $(3, \frac{13}{3})$ . If we wish, we can change the improper fractions to mixed numbers, $(1, 3 \frac{2}{3})$ , $(- 3, 2 \frac{1}{3})$ , $(3, 4 \frac{1}{3})$ .

A graph of a line passing through three points with coordinates negative three, seven over three; one, eleven over three; and three, thirteen over three.

$4 x + 4 y = 0$

We solve for $y$ .

$\begin{matrix} 4 y & = & - 4 x \\ y & = & - x \end{matrix}$

$x$	$y$	$(x, y)$
0	0	$(0, 0)$
2	$- 2$	$(2, - 2)$
$- 3$	3	$(- 3, 3)$

A graph of a line passing through three points with coordinates negative three, three; zero, zero; and two, negative two.

Notice that the $x -$ and $y -intercepts$ are the same point. Thus the intercept method does not provide enough information to construct this graph.

When an equation is given in the general form $a x + b y = c$ , usually the most efficient approach to constructing the graph is to use the intercept method, when it works.

Practice set b

Graph the following equations.

$x - 5 y = 5$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates zero, negative one and five, zero.

$x + 2 y = 6$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates zero, three and two, two.

$2 x + y = 1$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates zero, one and one, negative one.

Slanted, horizontal, and vertical lines

In all the graphs we have observed so far, the lines have been slanted. This will always be the case when both variables appear in the equation. If only one variable appears in the equation, then the line will be either vertical or horizontal. To see why, let’s consider a specific case:

Using the general form of a line, $a x + b y = c$ , we can produce an equation with exactly one variable by choosing $a = 0$ , $b = 5$ , and $c = 15$ . The equation $a x + b y = c$ then becomes

$0 x + 5 y = 15$

Since $0 \cdot (any number) = 0$ , the term $0 x$ is $0$ for any number that is chosen for $x$ .

Thus,

$0 x + 5 y = 15$

becomes

$0 + 5 y = 15$

But, $0$ is the additive identity and $0 + 5 y = 5 y$ .

$5 y = 15$

Then, solving for $y$ we get

$y = 3$

This is an equation in which exactly one variable appears.

This means that regardless of which number we choose for $x$ , the corresponding $y -value$ is 3. Since the $y -value$ is always the same as we move from left-to-right through the $x -values$ , the height of the line above the $x -axis$ is always the same (in this case, 3 units). This type of line must be horizontal.

An argument similar to the one above will show that if the only variable that appears is $x$ , we can expect to get a vertical line.

Sample set c

Graph $y = 4$ .
The only variable appearing is $y$ . Regardless of which $x -value$ we choose, the $y -value$ is always 4. All points with a $y -value$ of 4 satisfy the equation. Thus we get a horizontal line 4 unit above the $x -axis$ .

$x$	$y$	$(x, y)$
$- 3$	4	$(- 3, 4)$
$- 2$	4	$(- 2, 4)$
$- 1$	4	$(- 1, 4)$
0	4	$(0, 4)$
1	4	$(1, 4)$
2	4	$(2, 4)$
3	4	$(3, 4)$
4	4	$(4, 4)$

A graph of a line parallel to x-axis passing through a point with coordinates zero, four.

Graph $x = - 2$ .
The only variable that appears is $x$ . Regardless of which $y -value$ we choose, the $x -value$ will always be $- 2$ . Thus, we get a vertical line two units to the left of the $y -axis$ .

$x$	$y$	$(x, y)$
$- 2$	$- 4$	$(- 2, - 4)$
$- 2$	$- 3$	$(- 2, - 3)$
$- 2$	$- 2$	$(- 2, - 2)$
$- 2$	$- 1$	$(- 2, - 1)$
$- 2$	0	$(- 2, 0)$
$- 2$	1	$(- 2, 1)$
$- 2$	2	$(- 2, 0)$
$- 2$	3	$(- 2, 3)$
$- 2$	4	$(- 2, 4)$

Practice set c

Graph $y = 2$ .
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to x axis passing through a point with coordinates zero, two.

Graph $x = - 4$ .
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to y axis and passing through a point with coordinates negative four, zero.

Summarizing our results we can make the following observations:

When a linear equation in two variables is written in the form $a x + b y = c$ , we say it is written in general form .
To graph an equation in general form it is sometimes convenient to use the intercept method.
A linear equation in which both variables appear will graph as a slanted line.
A linear equation in which only one variable appears will graph as either a vertical or horizontal line.

$x = a$ graphs as a vertical line passing through $a$ on the $x -axis$ .
$y = b$ graphs as a horizontal line passing through $b$ on the $y -axis$ .

Exercises

For the following problems, graph the equations.

$- 3 x + y = - 1$
An xy-plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through three points with coordinates negative one, negative four; zero, negative one; and one, two.

$3 x - 2 y = 6$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$- 2 x + y = 4$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates negative two, zero and zero, four.

$x - 3 y = 5$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$2 x - 3 y = 6$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through three points with coordinates negative one, negative eight over three; zero, negative two; and three, zero.

$2 x + 5 y = 10$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$3 (x - y) = 9$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three, zero and zero, negative three.

$- 2 x + 3 y = - 12$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$y + x = 1$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through three points with coordinates one, zero; zero, one; and three, negative two.

$4 y - x - 12 = 0$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$2 x - y + 4 = 0$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates negative two, zero and zero, four.

$- 2 x + 5 y = 0$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$y - 5 x + 4 = 0$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through three points with coordinates zero, negative four; four over five, zero; and two, six.

$0 x + y = 3$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$0 x + 2 y = 2$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to x-axis and passing through two points with coordinates negative three, one and three, one.

$0 x + \frac{1}{4} y = 1$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$4 x + 0 y = 16$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to y-axis and passing through three points with coordinates four, zero; four, two; and four, four.

$\frac{1}{2} x + 0 y = - 1$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$\frac{2}{3} x + 0 y = 1$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$x = \frac{3}{2}$

A graph of a line parallel to y-axis in an xy plane. The line is labeled as ' x equals three over two' and it crosses the x-axis at x equals three over two.

$y = 3$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$y = - 2$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$y = - 2$

A graph of a line parallel to x-axis in an xy plane. The line is labeled as ' y equals negative two'. The line crosses the y-axis at y equals negative two.

$- 4 y = 20$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$x = - 4$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to y-axis in an xy plane. The line is labeled as ' x equals negative four'. The line crosses the x-axis at x equals negative four.

$- 3 x = - 9$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$- x + 4 = 0$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to y-axis in an xy plane. The line is labeled as ' x equals four'. The line crosses the x-axis at x equals four.

Construct the graph of all the points that have coordinates $(a, a)$ , that is, for each point, the $x -$ and $y -values$ are the same.
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

Calculator problems

$2.53 x + 4.77 y = 8.45$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three point three four, zero and zero, one point seven seven.

$1.96 x + 2.05 y = 6.55$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

$4.1 x - 6.6 y = 15.5$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three point seven eight, zero and zero, negative two point three five.

$626.01 x - 506.73 y = 2443.50$
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

Exercises for review

( [link] ) Name the property of real numbers that makes $4 + x = x + 4$ a true statement.

commutative property of addition

( [link] ) Supply the missing word. The absolute value of a number $a$ , denoted $| a |$ , is the from $a$ to $0$ on the number line.

( [link] ) Find the product $(3 x + 2) (x - 7)$ .

$3 x^{2} - 19 x - 14$

( [link] ) Solve the equation $3 [3 (x - 2) + 4 x] - 24 = 0$ .

( [link] ) Supply the missing word. The coordinate axes divide the plane into four equal regions called .

quadrants

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

tijani

what is titration

John Reply

what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

what is inorganic

emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

Ryan

what's motion

Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

Magreth

hello friend how are you

Muhammad Reply

fine, how about you?

Mohammed

Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

Reofrir Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra i for the community college' conversation and receive update notifications?

Ask

©flickr: Donnie	How well do you know Beanie Boos? By Angela Eckman Start Quiz
	Chemistry By OpenStax Read Online Course
	1 Physiotherapy Flashcards Set 1 By Rhodes Start Flashcards
	4 AP Key Terms 04 Tissue Level of Organization By OpenStax Start Key Terms
	Art Reviewer MCQ By Edgar Delgado Start Quiz
	2 Lec:2 Randomized Controlled Trials By Janet Forrester Start Quiz
©flickr: Miguel	Protozoal and Parasitic Infections By Cath Yu Start Quiz
	Anthropology Economic System By Richley Crapo Start Assignment
	13 AP 13 Nervous System MCQ By OpenStax Start Quiz
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz