Solving inequalities

Question

My teacher told me I can graph this with my graphingb calculator but I can't seem to get the correct inequalities for 1. X^4-10x^2+9<0

and 2.x^5+5x^4-4x^3+3x^2-2>0

Answer 1

#1 can be solved algebraically:

x⁴ -10x² + 9 ≤ 0

(x² - 9)(x² - 1) ≤ 0

(x - 3)(x + 3)(x - 1)(x + 1) ≤ 0

Thus, the key values are ±3, ±1

the intervals are ( -∞, -3), (-3, -1), (-1, 1), (1, 3), (3, ∞)

Next chose a value from each interval and substitute the value into the inequality in order to determine whether the value for that interval yields a positive or negative value. This process yields the solution:

[-3, -1] ∪ [1, 3].

This solution can be verified graphically.

#2 Graph 2x⁵ + 5x⁴ -4x³ + 3x² - 2

Use the zero function on the calculator to determine the x-intercepts. I used an x-window of [-10, 10] and a y-window of [-50, 50]

I got -3.24876, -0.55742, 0-74387

By inspecting the graph observe that the y-values in the interval (-3.24876, -0.55742) and (0.74387, ∞) are positive. Thus, these two intervals are the solutions.