7.1 Graphing linear equations and inequalities in one variable

Elementary algebra Page 1 / 1

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information.The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. Objectives of this module: understand the concept of a graph and the relationship between axes, coordinate systems, and dimension, be able to construct one-dimensional graphs.

Overview

Graphs
Axes, Coordinate Systems, and Dimension
Graphing in One Dimension

Graphs

We have, thus far in our study of algebra, developed and used several methods for obtaining solutions to linear equations in both one and two variables. Quite often it is helpful to obtain a picture of the solutions to an equation. These pictures are called graphs and they can reveal information that may not be evident from the equation alone.

The graph of an equation

The geometric representation (picture) of the solutions to an equation is called the graph of the equation.

Axes, coordinate systems, and dimension

Axis

The basic structure of the graph is the axis . It is with respect to the axis that all solutions to an equation are located. The most fundamental type of axis is the number line .

The number line is an axis

A number line with arrows on each end and is labeled from negative five to five in increments of one. There is an arrow pointing towards the number line with the label, 'This number line is an axis.'

We have the following general rules regarding axes.

Number of variables and number of axes

An equation in one variable requires one axis.
An equation in two variables requires two axes.
An equation in three variables requires three axes.
... An equation in $n$ variables requires $n$ axes.

We shall always draw an axis as a straight line, and if more than one axis is required, we shall draw them so they are all mutually perpendicular (the lines forming the axes will be at $90 °$ angles to one another).

Coordinate system

A system of axes constructed for graphing an equation is called a coordinate system .

The phrase, graphing an equation

The phrase graphing an equation is used frequently and should be interpreted as meaning geometrically locating the solutions to an equation.

Relating the number of variables and the number of axes

We will not start actually graphing equations until Section [link] , but in the following examples we will relate the number of variables in an equation to the number of axes in the coordinate system.

1. One-Dimensional Graphs
If we wish to graph the equation $5 x + 2 = 17$ , we would need to construct a coordinate system consisting of a single axis (a single number line) since the equation consists of only one variable. We label the axis with the variable that appears in the equation.

We might interpret an equation in one variable as giving information in one-dimensional space. Since we live in three-dimensional space, one-dimensional space might be hard to imagine. Objects in one-dimensional space would have only length, no width or depth.
2. Two-Dimensional Graphs
To graph an equation in two variables such as $y = 2 x - 3$ , we would need to construct a coordinate system consisting of two mutually perpendicular number lines (axes). We call the intersection of the two axes the origin and label it with a 0. The two axes are simply number lines; one drawn horizontally, one drawn vertically.

Recall that an equation in two variables requires a solution to be a pair of numbers. The solutions can be written as ordered pairs $(x, y)$ . Since the equation $y = 2 x - 3$ involves the variables $x$ and $y$ , we label one axis $x$ and the other axis $y$ . In mathematics it is customary to label the horizontal axis with the independent variable and the vertical axis with the dependent variable.

We might interpret equations in two variables as giving information in two-dimensional space. Objects in two-dimensional space would have length and width, but no depth.
3. Three-Dimensional Graphs
An equation in three variables, such as $3 x^{2} - 4 y^{2} + 5 z = 0$ , requires three mutually perpendicular axes, one for each variable. We would construct the following coordinate system and graph.

We might interpret equations in three variables as giving information about three-dimensional space.
4. Four-Dimensional Graphs
To graph an equation in four variables, such as $3 x - 2 y + 8 x - 5 w = - 7$ , would require four mutually perpendicular number lines. These graphs are left to the imagination.

We might interpret equations in four variables as giving information in four-dimensional space. Four-dimensional objects would have length, width, depth, and some other dimension.

Black holes
These other spaces are hard for us to imagine, but the existence of “black holes” makes the possibility of other universes of one-, two-, four-, or n -dimensions not entirely unlikely. Although it may be difficult for us “3-D” people to travel around in another dimensional space, at least we could be pretty sure that our mathematics would still work (since it is not restricted to only three dimensions)!

Graphing in one dimension

Graphing a linear equation in one variable involves solving the equation, then locating the solution on the axis (number line), and marking a point at this location. We have observed that graphs may reveal information that may not be evident from the original equation. The graphs of linear equations in one variable do not yield much, if any, information, but they serve as a foundation to graphs of higher dimension (graphs of two variables and three variables).

Sample set a

Graph the equation $3 x - 5 = 10$ .

Solve the equation for $x$ and construct an axis. Since there is only one variable, we need only one axis. Label the axis $x$ .

$\begin{array}{l} 3 x - 5 & = & 10 \\ 3 x & = & 15 \\ x & = & 5 \end{array}$

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Graph the equation $3 x + 4 + 7 x - 1 + 8 = 31$ .

Solving the equation we get,

$\begin{array}{l} 10 x + 11 & = & 31 \\ 10 x & = & 20 \\ x & = & 2 \end{array}$

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Practice set a

Graph the equation $4 x + 1 = - 7$ .

$x = - 2$

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Sample set b

Graph the linear inequality $4 x \geq 12$ .

We proceed by solving the inequality.

$\begin{array}{l} 4 x & \geq & 12 & Divide each side by 4 . \\ x & \geq & 3 \end{array}$

As we know, any value greater than or equal to 3 will satisfy the original inequality. Hence we have infinitely many solutions and, thus, infinitely many points to mark off on our graph.

The closed circle at 3 means that 3 is included as a solution. All the points beginning at 3 and in the direction of the arrow are solutions.

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Graph the linear inequality $- 2 y - 1 > 3$ .

We first solve the inequality.

$\begin{array}{l} - 2 y - 1 & > & 3 \\ - 2 y & > & 4 \\ y & < & - 2 & The inequality symbol reversed direction because we divided by –2 . \end{array}$

Thus, all numbers strictly less than $- 2$ will satisfy the inequality and are thus solutions.

Since $- 2$ itself is not to be included as a solution, we draw an open circle at $- 2$ . The solutions are to the left of $- 2$ so we draw an arrow pointing to the left of $- 2$ to denote the region of solutions.

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Graph the inequality $- 2 \leq y + 1 < 1$ .

We recognize this inequality as a compound inequality and solve it by subtracting 1 from all three parts.

$\begin{array}{l} - 2 \leq y + 1 < 1 \\ - 3 \leq y < 0 \end{array}$

Thus, the solution is all numbers between $- 3$ and 0, more precisely, all numbers greater than or equal to $- 3$ but strictly less than 0.

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Graph the linear equation $5 x = - 125$ .

The solution is $x = - 25$ . Scaling the axis by units of 5 rather than 1, we obtain

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Practice set b

Graph the inequality $3 x \leq 18$ .

$x \leq 6$

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Graph the inequality $- 3 m + 1 < 13$ .

$m > - 4$

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Graph the inequality $- 3 \leq x - 5 < 5$ .

$2 \leq x < 10$

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Graph the linear equation $- 6 y = 480$ .

$y = - 80$

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Exercises

For problems 1 - 25, graph the linear equations and inequalities.

$4 x + 7 = 19$

$x = 3$

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$8 x - 1 = 7$

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$2 x + 3 = 4$

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

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$x + 3 = 15$

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$6 y + 3 = y + 8$

$y = 1$

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$2 x = 0$

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$4 + 1 - 4 = 3 z$

$z = \frac{1}{3}$

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$x + \frac{1}{2} = \frac{4}{3}$

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$7 r = \frac{1}{4}$

$r = \frac{1}{28}$

A number line with arrows on each end, labeled from negative one over twenty-eight to three over twenty-eight in increments of one twenty-eighth. There is a closed circle at negative one over twenty-eight.

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$2 x - 6 = \frac{2}{5}$

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$x + 7 \leq 12$

$x \leq 5$

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$y - 5 < 3$

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$x + 19 > 2$

$x > - 17$

A number line with arrows on each end, labeled from negative twenty to zero, in increments of five. There is an open circle at negative seventeen. A dark arrow is originating from this circle, and heading towrads the right of negative seventeen.

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$z + 5 > 11$

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$3 m - 7 \leq 8$

$m \leq 5$

A number line with arrows on each end, labeled from negative one to eight, in increments of one. There is a closed circle at five. A dark arrow is originating from this circle, and heading towrads the left of five.

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$- 5 t \geq 10$

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$- 8 x - 6 \geq 34$

$x \leq - 5$

A number line with arrows on each end, labeled from negative eight to negative two, in increments of one. There is a closed circle at negative five. A dark arrow is originating from this circle, and heading towrads the left of negative five.

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$\frac{x}{4} < 2$

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$\frac{y}{7} \leq 3$

$y \leq 21$

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$\frac{2 y}{9} \geq 4$

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$\frac{- 5 y}{8} \leq 4$

$y \geq - \frac{32}{5}$

A number line with arrows on each end, labeled from negative seven to negative two, in increments of one. There is a closed circle at a point between negative six and negative seven labeled as negative thirty-two over five. A dark arrow is originating from this circle, and heading towrads the right of negative thirty-two.

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$\frac{- 6 a}{7} < - 4$

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$- 1 \leq x - 3 < 0$

$2 \leq x < 3$

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$6 \leq x + 4 \leq 7$

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$- 12 < - 2 x - 2 \leq - 8$

$3 \leq x < 5$

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Exercises for review

( [link] ) Simplify ${(3 x^{8} y^{2})}^{3}$ .

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( [link] ) List, if any should appear, the common factors in the expression $10 x^{4} - 15 x^{2} + 5 x^{6}$ .

$5 x^{2}$

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( [link] ) Solve the inequality $- 4 (x + 3) < - 3 x + 1$ .

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( [link] ) Solve the equation $y = - 5 x + 8 if x = - 2$ .

$(- 2, 18)$

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( [link] ) Solve the equation $2 y = 5 (3 x + 7) if x = - 1$ .

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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7.1 Graphing linear equations and inequalities in one variable

Overview

Graphs

The graph of an equation

Axes, coordinate systems, and dimension

Axis

The number line is an axis

Number of variables and number of axes

Coordinate system

The phrase, graphing an equation

Relating the number of variables and the number of axes

Black holes

Graphing in one dimension

Sample set a

Practice set a

Sample set b

Practice set b

Exercises

Exercises for review

Questions & Answers

Read also:

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