▪ Solving Inequalities Using Graphs of Functions

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
-3/4x+3 is solid line, -3/4x+3>0 shaded with dotted line at x=4 and y=-3/4x+3 when x<4 is solid line Key: -3/4x+3 is solid line, -3/4x+3>0 shaded with dotted line at x=4 and y=-3/4x+3 when x<4 is solid line 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
Graphs: -3/4x+3 is solid line, -3/4x+3<0 shaded with dotted line at x=4 and y=-3/4x+3 when x>4 is solid line Key: -3/4x+3 is solid line, -3/4x+3<0 shaded with dotted line at x=4 and y=-3/4x+3 when x>4 is solid line 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.

x3x4x2+40

Try It

Solve the inequality graphically. Use a graphing tool.

(x1)43(x1)240

Try It

Solve the inequality graphically. Use a graphing tool.

(x+1)234>0

Solving Equations with One Variable using Multiple Functions

Example: Solving f(x)>g(x) Graphically

Solve the inequality graphically. Use a graphing tool.

2x23>2x+1

Try It

Solve the inequality graphically. Use a graphing tool.

2x3+6x3x23