Learning Outcomes
- Solve inequalities with one variable using graphs of functions.
Solving Inequalities with One Variable using One Function
Recall DISTINCT PARTS OF A GRAPH of a Function AND THEIR y VALUES
(a) When a graph is below the x-axis, y values are negative. So, f(x)<0.
(b) When a graph is on the x-axis, y values are zero. So, f(x)=0.
(c) When a graph is above the x-axis, y values are positive. So, f(x)>0.
As we have seen, we can solve an equation f(x)=0 by finding the x-intercepts of its function y=f(x). That means we can solve an equation f(x)=0 by finding the x values when the graph of y=f(x) is on the x-axis. We can apply the idea to solve inequalities with one variable.
In the previous section, we talked about −34x+3=0. Now let’s consider the inequalities −34x+3>0 and −34x+3<0. According to the relationship above, to solve −34x+3>0, we can find the x values when the graph of f(x)=−34x+3 is above the x-axis and below the x-axis, respectively.
Inequality |
Position |
Graph |
|
Solution |
−34+3>0 |
above
the x-axis |
|
|
When the graph is above the x-axis, which is the orange part,
(−∞,4) or {x|x<4}
|
−34+3<0 |
below
the x-axis |
|
|
When the graph is below the x-axis, which is the green part,
(4,∞) or {x|x>4} |
Example: Solving f(x)≥0 Graphically
Solve the inequality graphically. Use a graphing tool.
x3−x−4x2+4≥0
Show Answer
Let f(x)=x3−x−4x2+4. Then find the parts that are above or on the x-axis because f(x) is “greater than (>),” which is above the x-axis, or “equal to (=),” which is on the x-axis. From the graph, we can see that the graph of f(x)=x3−x−4x2+4 is above the x-axis when [latex]-14[/latex] and on the x-axis when x=−1,1,4. So, the solution of the inequality x3−x−4x2+4≤0 is [−1,1]∪[4,∞) or {x|−1≤x≤1 or x≥4}.
Try It
Solve the inequality graphically. Use a graphing tool.
(x−1)4−3(x−1)2−4≤0
Show Answer
Let f(x)=(x−1)4−3(x−1)2−4. Then find the parts that are below or on the x-axis. From the graph, we can see that the graph of f(x)=(x−1)4−3(x−1)2−4 is below the x-axis when [latex]-1
Try It
Solve the inequality graphically. Use a graphing tool.
3√(x+1)2−4>0
Show Answer
Let f(x)=3√(x+1)2−4. Then find the parts that are above the x-axis. From the graph, we can see that the graph of f(x)=3√(x+1)2−4 is above the x-axis when x<−9 or x>7. So, the solution of the inequality 3√(x+1)2>4 is (−∞,−9)∪(7,∞) or {x|x<−9 or x>7}.
Solving Equations with One Variable using Multiple Functions
Example: Solving f(x)>g(x) Graphically
Solve the inequality graphically. Use a graphing tool.
2x2−3>−2x+1
Show Answer
Method 1
Move all terms to the left: 2x2−3+2x−1>0*
* If you are using a graphing tool, you don’t need to simplify the inequality. Just let f(x)=2x2−3−(−2x+1) or f(x)=2x2−3+2x−1.
So, 2x2+2x−4>0.
Let f(x)=2x2+2x−4. Then find the parts that are above the x-axis.
Since the graph of f(x)=2x2+2x−4 is above the x-axis when x<−2 or x>1, the solution of the inequality 2x2−3>−2x+1 is (−∞,−2)∪(1,∞) or {x|x<−2 or x>1}.
Method 2
Let f(x)=2x2−3 and g(x)=−2x+1. Then find the parts where f(x)>g(x), which means where the graph of f(x) is above the graph of g(x).
Since the graph of f(x)=2x2−3 is above the graph of g(x)=−2x+1 when x<−2 or x>1, the solution of the inequality 2x2−3>−2x+1 is (−∞,−2)∪(1,∞) or {x|x<−2 or x>1}.
Try It
Solve the inequality graphically. Use a graphing tool.
−2x3+6x−3≤x2−3
Show Answer
Move all terms to the lefthand side: −2x3+6x−3−x2+3≤0. Then let f(x)=−2x3−x2+6x. Since the graph off(x)=−2x3−x2+6x is below the x-axis when [latex]-2\frac{3}{2}[/latex] and on the x-axis when x=−2,0,32, the solution of the inequality −2x3+6x−3≤x2−3 is [−2,0]∪[32,∞) or {x|−2≤x≤0 or x≥32}.
OR
Let g(x)=−2x3+6x−3 and h(x)=x2−3. Since the graph of g(x)=−2x3+6x−3 is below the graph of h(x)=x2−3 when [latex]-2\frac{3}{2}[/latex] and intersects the graph of h(x)=x2−3 when x=−2,0,32, the solution of the inequality −2x3+6x−3≤x2−3 is [−2,0]∪[32,∞) or {x|−2≤x≤0 or x≥32}.
Candela Citations
CC licensed content, Original
- Solving Inequalities Using Graphs of Functions. Authored by: Michelle Eunhee Chung. Provided by: Georgia State University. License: CC BY: Attribution