You can solve inequalities of the form a x 2 + b x + c > 0 eh , x squared , plus b x plus c greater than 0  or a x 2 + b x + c < 0 eh , x squared , plus b x plus c less than 0  by graphing the corresponding function and seeing where the graph is above or below the x-axis.

Example 2

Find the solution sets for 1 4 ( x − 2 ) 2 − 1 > 0  and  1 4 ( x − 2 ) 2 − 1 < 0 . 1 fourth , open bold italic x minus 2 , close squared , minus 1 greater than 0 , and , 1 fourth , open bold italic x minus 2 , close squared , minus 1 less than 0 .

The solution set for 1 4 ( x − 2 ) 2 − 1 > 0 1 fourth , open x minus 2 , close squared , minus 1 greater than 0  is all x-values of points on the parabola that lie above the x-axis.

x < 0 x less than , 0  or x > 4 x greater than 4

The solution set for 1 4 ( x − 2 ) 2 − 1 < 0 1 fourth , open x minus 2 , close squared , minus 1 less than 0  is all x-values of points on the parabola that lie below the x-axis.

0 < x < 4 0 less than x less than 4

Example 3

Solve − 2 x 2 − 8 x − 6 < 0 . negative 2 , bold italic x squared , minus 8 bold italic x minus 6 less than 0 .

Think: Since the coefficient of x 2 x squared  is less than zero, the graph of y = − 2 x 2 − 8 x − 6 y equals negative 2 , x squared , minus 8 x minus 6  opens downward.

Solve: Find where − 2 x 2 − 8 x − 6 negative 2 , x squared , minus 8 x minus 6  equals 0.

− 2 x 2 − 8 x − 6 = 0 − 2 ( x 2 + 4 x + 3 ) = 0 − 2 ( x + 3 ) ( x + 1 ) = 0 x = − 3  or  x = − 1   table with 4 rows and 2 columns , row1 column 1 , negative 2 , x squared , minus 8 x minus 6 , column 2 equals 0 , row2 column 1 , negative 2 . open . x squared , plus 4 x plus 3 . close , column 2 equals 0 , row3 column 1 , negative 2 . open , x plus 3 , close . open , x plus 1 , close , column 2 equals 0 , row4 column 1 , x equals negative 3 , or , x equals negative 1 , column 2 , end table

The graph of y = − 2 x 2 − 8 x − 6 y equals negative 2 , x squared , minus 8 x minus 6  opens down and crosses the x-axis at x = − 3 x equals negative 3  and x = − 1 . x equals negative 1 .  The solution of − 2 x 2 − 8 x − 6 < 0 negative 2 , x squared , minus 8 x minus 6 less than 0  is x < − 3  or  x > − 1 . x less than negative 3 , or , x greater than negative 1 .

Exercises

5. Solve each inequality. Graph your solution on a number line.
   1. x 2 < 36 x squared , less than 36
   2. x 2 − 9 > 0 x squared , minus 9 greater than 0
   3. x 2 < − 4 x squared , less than negative 4
   4. x 2 − 3 x − 18 > 0 x squared , minus 3 x minus 18 greater than 0
6. How can you use the graph of y = 3 x − 4 y equals 3 , x minus , 4  to solve the linear inequality 3 x − 4 < 0 ? 3 x minus 4 less than 0 question mark  Graph the solution.
7. How can you solve the absolute value inequality | − 3 x + 4 | > 0 ? vertical line negative 3 x plus 4 vertical line greater than 0 question mark
8. Example 2 shows two possible graphs for a quadratic inequality. What other possibilities are there?
9. Describe the graphs of possible solutions of a x 3 + b x 2 + c x + d > 0 . eh , x cubed , plus b , x squared , plus c x plus d greater than 0 .

Table of Contents