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Graphing Linear Equations and Inequalities in One Variable

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 nn variables requires nn 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° 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 (Reference), 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 5x+2=17 5x+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.

    A number line with arrows on each side labeled from negative four to six in increments of one. The number line is labeled x. There is an arrow pointing towards the number line with the label, 'This axis is a line and lines are one dimensional length.' There is another arrow pointing towards the number three with a label, 'Graphs are points.'

    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=2x3 y=2x3 , 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.

    An xy plane with gridlines and a straight line passing through quadrants one, three, and four. There is an arrow pointing towards this line with the label 'In general, graphs are curves (straight or curved).' There is another arrow pointing towards the xy plane with the label 'This is a plane and planes are two-dimensional: length and width.'

    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) (x,y) . Since the equation y=2x3 y=2x3 involves the variables xx and yy, we label one axis xx and the other axis yy. 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 +5z=0 3 x 2 4 y 2 +5z=0 , requires three mutually perpendicular axes, one for each variable. We would construct the following coordinate system and graph.

    A three dimensional x y z plane and a graph of an arbitrary surface. There are arrows pointing towards the surface with the following labels: ‘This is three dimensional: length, width, and depth. Graphs are surfaces. We won't consider these types of graphs as they are too complicated to draw.’

    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 3x2y+8x5w=7 3x2y+8x5w=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

Example 1

Graph the equation 3x5=10 3x5=10 .

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

3x5 = 10 3x = 15 x = 5 3x5 = 10 3x = 15 x = 5
A number line labeled x with arrows on each end, labeled from negative three to seven, in increments of one. There is a closed circle on five.

Example 2

Graph the equation 3x+4+7x1+8=31 3x+4+7x1+8=31 .

Solving the equation we get,

10x+11 = 31 10x = 20 x = 2 10x+11 = 31 10x = 20 x = 2
A number line labeled x with arrows on each end, labeled from negative five to five, in increments of one. There is a closed circle on two.

Practice Set A

Exercise 1

Graph the equation 4x+1=7 4x+1=7 .
A horizontal line with arrows on both ends.

Solution

x=2 x=2
A number line labeled x with arrows on each end, labeled from negative five to five, in increments of one. There is a closed circle on negative two.

Sample Set B

Example 3

Graph the linear inequality 4x 12 4x 12 .

We proceed by solving the inequality.

4x 12 Divide each side by 4. x 3 4x 12 Divide each side by 4. x 3

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.
A number line labeled x with arrows on each end, labeled from negative four to six, in increments of one. There is a closed circle on three with a dark shaded arrow to the right of three.

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.

Example 4

Graph the linear inequality 2y1>3 2y1>3 .

We first solve the inequality.

2y1 > 3 2y > 4 y < 2 The inequality symbol reversed direction because we divided by –2. 2y1 > 3 2y > 4 y < 2 The inequality symbol reversed direction because we divided by –2.

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

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

A number line labeled y with arrows on each end, labeled from negative five to five, in increments of one. There is a open circle on negative two with a dark shaded arrow to the left of negative two.

Example 5

Graph the inequality 2 y+1<1 2 y+1<1 .

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

2 y+1<1 3 y<0 2 y+1<1 3 y<0

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

A number line labeled y with arrows on each end, and labeled from negative five to five in increments of one. There is a closed circle at negative three and an open circle at zero, with a black shaded line connecting the two circles.

Example 6

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

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

A number line labeled x with arrows on each end, and labeled from negative fifty to fifty in increments of ten. There is a closed circle labeled negative twenty five, halfway between negative thirty and negative twenty.

Practice Set B

Exercise 2

Graph the inequality 3x18 3x18 .
A horizontal line with arrows on both ends.

Solution

x6 x6
A number line labeled x with arrows on each end, labeled from negative two to eight, in increments of one. There is a closed circle on six with a dark shaded arrow to the left of six.

Exercise 3

Graph the inequality 3m+1<13 3m+1<13 .
A horizontal line with arrows on both ends.

Solution

m>4 m>4
A number line labeled m with arrows on each end, labeled from negative five to five, in increments of one. There is a open circle on negative four with a dark shaded arrow to the right of negative four.

Exercise 4

Graph the inequality 3 x5<5 3 x5<5 .
A horizontal line with arrows on both ends.

Solution

2 x<10 2 x<10
A number line labeled x with arrows on each end, and labeled from one to eleven in increments of one. There is a closed circle at two and an open circle at ten, with a black shaded line connecting the two circles.

Exercise 5

Graph the linear equation 6y=480 6y=480 .
A horizontal line with arrows on both ends.

Solution

y=80 y=80
A number line labeled y with arrows on each end, labeled from negative hundred to zero, in increments of ten. There is a closed circle on negative eighty.

Exercises

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

Exercise 6

4x+7=19 4x+7=19
A horizontal line with arrows on both ends.

Solution

x=3 x=3

A number line with arrows on each end, labeled from negative two to four in increments of one. There is a closed circle at three.

Exercise 7

8x1=7 8x1=7
A horizontal line with arrows on both ends.

Exercise 8

2x+3=4 2x+3=4
A horizontal line with arrows on both ends.

Solution

x= 1 2 x= 1 2

A number line with arrows on each end, labeled from negative two to four in increments of one. There is a closed circle at a point between zero and one.

Exercise 9

x+3=15 x+3=15
A horizontal line with arrows on both ends.

Exercise 10

6y+3=y+8 6y+3=y+8
A horizontal line with arrows on both ends.

Solution

y=1 y=1

A number line with arrows on each end, labeled from negative two to four in increments of one. There is a closed circle at one.

Exercise 11

2x=0 2x=0
A horizontal line with arrows on both ends.

Exercise 12

4+14=3z 4+14=3z
A horizontal line with arrows on both ends.

Solution

z= 1 3 z= 1 3

A number line with arrows on each end, labeled from negative one to two in increments of one third. There is a closed circle at one third.

Exercise 13

x+ 1 2 = 4 3 x+ 1 2 = 4 3
A horizontal line with arrows on both ends.

Exercise 14

7r= 1 4 7r= 1 4
A horizontal line with arrows on both ends.

Solution

r= 1 28 r= 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.

Exercise 15

2x6= 2 5 2x6= 2 5
A horizontal line with arrows on both ends.

Exercise 16

x+712 x+712
A horizontal line with arrows on both ends.

Solution

x5 x5

A number line with arrows on each end, labeled from negative one to six, 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.

Exercise 17

y5<3 y5<3
A horizontal line with arrows on both ends.

Exercise 18

x+19>2 x+19>2
A horizontal line with arrows on both ends.

Solution

x>17 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.

Exercise 19

z+5>11 z+5>11
A horizontal line with arrows on both ends.

Exercise 20

3m78 3m78
A horizontal line with arrows on both ends.

Solution

m5 m5

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.

Exercise 21

5t10 5t10
A horizontal line with arrows on both ends.

Exercise 22

8x634 8x634
A horizontal line with arrows on both ends.

Solution

x5 x5

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.

Exercise 23

x 4 <2 x 4 <2
A horizontal line with arrows on both ends.

Exercise 24

y 7 3 y 7 3
A horizontal line with arrows on both ends.

Solution

y21 y21

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

Exercise 25

2y 9 4 2y 9 4
A horizontal line with arrows on both ends.

Exercise 26

5y 8 4 5y 8 4
A horizontal line with arrows on both ends.

Solution

y 32 5 y 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.

Exercise 27

6a 7 <4 6a 7 <4
A horizontal line with arrows on both ends.

Exercise 28

1x3<0 1x3<0
A horizontal line with arrows on both ends.

Solution

2x<3 2x<3

A number line with arrows on each end, labeled from zero to four in increments of one. There is a closed circle at two, and an open circle at three. These circles are connected by a black line.

Exercise 29

6x+47 6x+47
A horizontal line with arrows on both ends.

Exercise 30

12<2x28 12<2x28
A horizontal line with arrows on both ends.

Solution

3x<5 3x<5

A number line with arrows on each end, labeled from two to five in increments of one. There is a closed circle at three and an open circle at five. These circles are connected by a black line.

Exercises for Review

Exercise 31

((Reference)) Simplify (3 x 8 y 2 ) 3 (3 x 8 y 2 ) 3 .

Exercise 32

((Reference)) List, if any should appear, the common factors in the expression 10 x 4 15 x 2 +5 x 6 10 x 4 15 x 2 +5 x 6 .

Solution

5 x 2 5 x 2

Exercise 33

((Reference)) Solve the inequality 4(x+3)<3x+1 4(x+3)<3x+1 .

Exercise 34

((Reference)) Solve the equation y=5x+8ifx=2 y=5x+8ifx=2 .

Solution

( 2,18 ) ( 2,18 )

Exercise 35

((Reference)) Solve the equation 2y=5(3x+7)ifx=1 2y=5(3x+7)ifx=1 .

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