Solve the quadratic inequality by locating the x-intercept(s) (if they exist), and noting the end behavior of the graph. Begin by writing the inequality in function form as needed.

Question

Solve the quadratic inequality by locating the x-intercept(s) (if they exist), and noting the end behavior of the graph. Begin by writing the inequality in function form as needed.



Answer 1

5x² ≥ 4x - 4

First, write the inequality in function form:

f(x) ≥ 5x² - 4x + 4            [ f(x) ≥ 0 ]

Now, use the quadratic formula:        x = ( -b ± √(b² -4ac) ) / (2a)

1)      x = ( 4 + √(16 – 80) ) / 40

x = ( 4 + √(-8) ) / 40

or

2) x = ( 4 - √(-8) ) / 40

Locate the x-intercept(s)   [the value of x that produces f(x)=0 ]

      There are no x-intercept(s).

Note the end behavior:

    f(x) ≥ 0    is true for every x

5x² ≥ 4x - 4  is true for every x

The solution is -∞ < x < ∞ (or "all real numbers").

Answer 2

The inequality does not have any zeros.

The inequality is true for all values of x.

Answer 3

The 5x² can be graphed as y = 5x² like this:

The "4x - 4" can be graphed as y = 4x - 4 like this:

Now, look for all the places where the y = 5x² (the red line) is greater than or equal to the y = 4x - 4 (the blue line) and you see it is everywhere so the solution is -∞ < x < ∞ (or "all real numbers")