Х²-5х+4<0
To solve the inequality x² - 5x + 4 < 0, you can follow these steps:
First, factor the quadratic equation if possible. In this case, the equation x² - 5x + 4 cannot be factored easily, so we'll use an alternative method.
Set the equation equal to zero to find the x-intercepts. x² - 5x + 4 = 0
Use the quadratic formula to find the roots of the equation: x = (-b ± √(b² - 4ac)) / (2a)
In this equation, a = 1, b = -5, and c = 4. Substituting these values: x = (-(-5) ± √((-5)² - 4(1)(4))) / (2(1)) x = (5 ± √(25 - 16)) / 2 x = (5 ± √9) / 2 x = (5 ± 3) / 2 x1 = (5 + 3) / 2 = 4 x2 = (5 - 3) / 2 = 1
The x-intercepts are x = 1 and x = 4.
Graph the equation x² - 5x + 4 on a number line, marking the x-intercepts.
We have the x-intercepts at x = 1 and x = 4. Mark these points on the number line.
Number line: ------1-------4------
Test intervals between the x-intercepts.
Choose a value within each interval and substitute it into the inequality to check if it holds true or false.
Let's test the intervals: (-∞, 1), (1, 4), (4, ∞)
For x = 0 (within the interval (-∞, 1)), substitute it into the inequality: 0² - 5(0) + 4 < 0 4 < 0 This is false.
For x = 2 (within the interval (1, 4)), substitute it into the inequality: 2² - 5(2) + 4 < 0 0 < 0 This is also false.
For x = 5 (within the interval (4, ∞)), substitute it into the inequality: 5² - 5(5) + 4 < 0 4 < 0 This is false as well.
Analyze the results.
Based on the tests, none of the intervals satisfy the inequality x² - 5x + 4 < 0. Therefore, the solution to the inequality is an empty set or ∅.
In conclusion, the inequality x² - 5x + 4 < 0 has no solution.
