solve rational inequality

Question

Solve the rational inequality and write the solution in interval notation 2 − 5/𝑥 + 2/ 𝑥 ^2 ≥ 0

Answer 1

The function does not exist at x =0. Thus, multiplying by x² for x ≠ 0 will preserve the inequality.

2x^2 - 5x + 2 >= 0

(2x - 1/2)(x - 2) >= 0

0s are at 2 and 1/2

This will be greater than or equal to 0 for x >= 2 and x <= 1/2 with the exception of x = 0

(-∞, 0) υ (0, 1/2] υ [2, ∞)

Answer 2

Jade

Given an inequality involving fractions multiply through by something to clear the fractions.

Here we can multiply by x^2, giving 2x²-5x+2>=0. (x² has to be zero or positive, so the inequality sign remains the same. If we were multiplying through by a negative quantity, we would reverse the inequality sign).

Now we solve 2x²-5x+2 = 0. This factorises as (2x-1) (x-2) = 0, giving roots x=0.5 (from 2x-1 = 0) and x=2 (from x-2=0).

As the coefficient of x² is positive, the graph is concave up, meaning the function will be positive or zero before the lowest root or above the highest root.

So the function will be >=0 when x<=0.5 or x >=2

In interval notation this can be written as (-∞,0.5] ∪ [2,+∞)

The first interval contains all real numbers <=0.5. The interval is open on the left , and closed on the right as the point x=0.5 needs to be included

The second interval contains all real numbers >=2. The interval is open on the right, and closed on the left as the point x=2 needs to be included.

Mike