e Think about what you need in order to graph an inequality using the previous parts.
A
aTable:
Point
f(x)≥ x-1
True/False
f(x)≤ x-1
True/False
(-4,2)
f(-4)≥ -4-1
true
f(-4)≤ -4-1
false
(-2,2)
f(-2)≥ -2-1
true
f(-2)≤ -2-1
false
(0,2)
f(0)≥ 0-1
true
f(0)≤ 0-1
false
(2,2)
f(2)≥ 2-1
true
f(2)≤ 2-1
false
(4,2)
f(4)≥ 4-1
false
f(4)≤ 4-1
true
B
bGraph:
C
cGraph:
D
dConjectures: See solution.
Table:
Point
f(x)≥ x-1
True/False
f(x)≤ x-1
True/False
(-4,2)
f(-4)≥ -4-1
true
f(-4)≤ -4-1
false
(-2,2)
f(-2)≥ -2-1
true
f(-2)≤ -2-1
false
(0,2)
f(0)≥ 0-1
true
f(0)≤ 0-1
false
(2,2)
f(2)≥ 2-1
true
f(2)≤ 2-1
false
(4,2)
f(4)≥ 4-1
false
f(4)≤ 4-1
true
(1,1)
f(1)≥ 1-1
true
f(1)≤ 1-1
false
(1,-1)
f(1)≥ 1-1
false
f(1)≤ 1-1
true
(1,0)
f(1)≥ 1-1
true
f(1)≤ 1-1
true
E
e See solution.
Practice makes perfect
a In this exercise, we will investigate the graph of linear inequalities on the coordinate plane. Let's complete the given table. Remember that f(x) is just another way to refer to the y-values of a function.
f(x) & = mx + b
f(x) &= 1 x + ( - 1)
In this case, the slope of the function is 1 and the y-intercept is -1. We will plot the y-intercept and find a second point in order to draw the line.
Now, we will connect the two points and draw the line.
c Let's plot the points from Part A. The ones that make f(x)≥ x-1 true will be red and the one that makes f(x)≤ x-1 true will be blue.
d Let's think about the implications of our findings from Parts A and C and make conjectures about the graphs of the inequalities.
f(x)≥ x-1: all of the points above the line will be included in the solution set.
f(x)≤ x-1: all of the points below the line will be included in the solution set.
Let's check three points: one above the line, one below the line, and one on the line.
Point
f(x)≥ x-1
True/False
f(x)≤ x-1
True/False
(-4,2)
f(-4)≥ -4-1
true
f(-4)≤ -4-1
false
(-2,2)
f(-2)≥ -2-1
true
f(-2)≤ -2-1
false
(0,2)
f(0)≥ 0-1
true
f(0)≤ 0-1
false
(2,2)
f(2)≥ 2-1
true
f(2)≤ 2-1
false
(4,2)
f(4)≥ 4-1
false
f(4)≤ 4-1
true
(1,1)
f(1)≥ 1-1
true
f(1)≤ 1-1
false
(1,-1)
f(1)≥ 1-1
false
f(1)≤ 1-1
true
(1,0)
f(1)≥ 1-1
true
f(1)≤ 1-1
true
As we can see, (1,1) satisfies f(x)≥ x-1, (1,-1) satisfies f(x)≤ x-1, and (1,0) satisfies both.
e If a point satisfies an inequality, all points on the coordinate plane on that side of the line will satisfy the inequality.
If a point does not satisfy an inequality, none of the points on that side of the line will satisfy the inequality.
