Х²-5х+4<0

To solve the inequality х²-5х+4<0, we can follow these steps:

Step 1: Factorize the quadratic equation х²-5х+4 = 0. We can find two numbers whose product is 4 and whose sum is -5. The numbers are -1 and -4. So, х²-5х+4 can be factored as (х-1)(х-4) = 0.

Step 2: Solve the equation (х-1)(х-4) = 0. Setting each factor equal to zero: х - 1 = 0 х = 1

х - 4 = 0 х = 4

So, the solutions of the equation are х = 1 and х = 4.

Step 3: Plot the solutions on the number line.

|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____

 -∞    1     4     +∞

Step 4: Test a value from each interval to determine whether it satisfies the inequality х²-5х+4 < 0.

Testing the interval (-∞, 1): For example, let's choose х = 0. Substituting into the inequality, we get: 0² - 5(0) + 4 = 4 > 0. Since it's not less than zero, this interval is not part of the solution.

Testing the interval (1, 4): For example, let's choose х = 3. Substituting into the inequality, we get: 3² - 5(3) + 4 = 4 - 15 + 4 = -7 < 0. Since it satisfies the inequality, this interval is part of the solution.

Testing the interval (4, +∞): For example, let's choose х = 5. Substituting into the inequality, we get: 5² - 5(5) + 4 = 25 - 25 + 4 = 4 > 0. Since it's not less than zero, this interval is not part of the solution.

Step 5: Write the solution in interval notation. The solution to the inequality х²-5х+4<0 is (1, 4).