what is the solution of this inequality?: x+3/x^2-5x+4>_0?
To find the solution to the inequality x + 3 / (x^2 - 5x + 4) ≥ 0, we can follow these steps:
Step 1: Factorize the denominator (x^2 - 5x + 4):
The denominator can be factored as (x - 1)(x - 4).
Step 2: Determine the critical values:
Set the denominator equal to zero and solve: (x - 1)(x - 4) = 0.
This yields two critical values: x = 1 and x = 4.
Step 3: Create a number line:
Plot the critical values on a number line, dividing it into three intervals: (-∞, 1), (1, 4), and (4, +∞).
Step 4: Choose a test point in each interval:
To determine if the inequality holds in each interval, we select test points that are not critical values. Let's choose -2, 2, and 5 as our test points (you can choose other numbers if you prefer).
Step 5: Evaluate the inequality for the test points:
For x = -2: Substitute this value into the inequality: (-2 + 3) / ((-2)^2 - 5(-2) + 4) = 1 / 14. As 1/14 > 0, the inequality holds for x < 1.
For x = 2: Substitute this value into the inequality: (2 + 3) / (2^2 - 5(2) + 4) = 5 / 2. As 5/2 > 0, the inequality holds for 1 < x < 4.
For x = 5: Substitute this value into the inequality: (5 + 3) / (5^2 - 5(5) + 4) = 8 / 14. As 8/14 > 0, the inequality holds for x > 4.
Step 6: Determine the solution:
Based on the number line and the test points, we see that the inequality holds for x in the intervals (-∞, 1), (1, 4), and (4, +∞).
Therefore, the solution to the inequality x + 3 / (x^2 - 5x + 4) ≥ 0 is:
x < 1 or 1 < x < 4 or x > 4.