solve the inequality express your solution using interval notation
x^4-7x^3+12x^2<0
y = x^2 * (x^2 - 7x + 12)
= x^2 (x-3)(x-4)
x^2 is always positive, so (x-3)(x-4) must be negative. That is true only for
3 < x < 4 or x in (3,4)
x^2(x^2 - 7x + 12) < 0
x^2(x-3)(x-4) < 0
critical values are x = 0, 3, 4
test for x< 0 , say x = -1
all we care about is the sign, so
(+)(-)(-) < 0 false
test for x between 0 and 3, says x = 1
(+)(-)(-) < 0 false
test for x between 3 and 4, say x = 3.5
(+)(+)(-) < 0 ---> TRUE
test for x > 4 , say x = 5
(+)(+)(+) < 0 false
so the solution is
3 < x < 4
the graph shown here, confirms this
http://www.wolframalpha.com/input/?i=x%5E4-7x%5E3%2B12x%5E2
http://www.wolframalpha.com/input/?i=x%5E4-7x%5E3%2B12x%5E2
To solve the inequality x^4 - 7x^3 + 12x^2 < 0, we need to find the values of x that make the inequality true.
Step 1: Factor the expression on the left side of the inequality if possible.
x^4 - 7x^3 + 12x^2 can be factored as x^2(x^2 - 7x + 12).
Step 2: Set each factor equal to zero and solve for x.
x^2 = 0 or x^2 - 7x + 12 = 0.
For x^2 = 0, taking the square root of both sides gives x = 0.
For x^2 - 7x + 12 = 0, we can factor the quadratic equation to (x - 3)(x - 4) = 0. Setting each factor equal to zero gives x = 3 or x = 4.
Step 3: Use a sign chart to determine the intervals where the expression is positive or negative.
To construct the sign chart, we'll evaluate the expression x^4 - 7x^3 + 12x^2 for x-values less than 0, between 0 and 3, between 3 and 4, and greater than 4.
For x < 0, we can choose x = -1. Plugging this value into the expression gives (-1)^4 - 7(-1)^3 + 12(-1)^2 = 1 + 7 + 12 = 20 > 0.
For 0 < x < 3, we can choose x = 1. Plugging this value into the expression gives (1)^4 - 7(1)^3 + 12(1)^2 = 1 - 7 + 12 = 6 > 0.
For 3 < x < 4, we can choose x = 3.5. Plugging this value into the expression gives (3.5)^4 - 7(3.5)^3 + 12(3.5)^2 = 85.375 - 451.125 + 178.5 = -187.25 < 0.
For x > 4, we can choose x = 5. Plugging this value into the expression gives (5)^4 - 7(5)^3 + 12(5)^2 = 625 - 875 + 300 = 50 > 0.
Step 4: Write the solution using interval notation.
From the sign chart, we can see that the expression x^4 - 7x^3 + 12x^2 < 0 when x belongs to the interval (3, 4).
So, the solution to the inequality is x ∈ (3, 4) in interval notation.