b Plot y≤ 4. Which parts of the graph correspond to the inequality?
C
c Where on the graph is f(x)=3?
D
d Plot y> 2. Which parts of the graph correspond to the inequality?
A
a x=3
B
b 0≤ x ≤ 6
C
c x=1, x=5
D
d x<2, x>4
Practice makes perfect
a The graph describes the function expression |x-3|+1. To find when it's equal to 1 we should locate the point(s) on the graph where y=1.
The function value is 1 when x=3. The solution to the equation is therefore x=3.
b The solution set to the inequality is all x-values that make the function value less than or equal to 4. In the coordinate plane, let's draw the line y=4.
The x-values of every point on the graph that are beneath or equal to y=4 solves the inequality.
The points on the graph beneath y=4 are between x=0 and x=6 including the endpoints. We can write this as the following solution set.
0≤ x ≤ 6
c Now we want to find for which x-values |x-3|+1 is equal to 3. Start from y=3 on the y-axis and determine the point(s) on the graph that have this y-value.
The function value is 3 when x=1 and x=5. Therefore, the solutions to the equation are
x=1 and x=5.
d Now we want to find all x-values where the function's value is greater than 2. Let's start by drawing the line y=2. Notice that we will dash the line since we are dealing with a strict inequality. This means any x that gives y=2 will not be included in the solution set.
The x-values of every point on the graph that is above y=2 solves the inequality.
The parts of the graph that satisfies the inequality are x less than 2 or x greater than 4. The solution set can be described with the following intervals
x<2 and x>4
