solve the inequality using any method
x^2-8x+12<0
Princess, Key, or whoever you are -- it's disconcerting to see three similar problems in a row -- all from the same computer. Please stick with one name so that our tutors can help you better.
x^2-8x+12<0
(x-6)(x-2) < 0
so we have "critical" values at x=6 and x=2
mark these on a number line, yielding 3 sections
x<2, test at x=0 in (x-6)(x-2)<0, no!
x between 2 and 6 , test at x=5, yes!
x > 6 , test at x=10, no!
so 2 < x < 6
thank you reiny for helping me and my causins were useing my computer for help on there math homework also!!! so thank you from all of us
To solve the inequality x^2 - 8x + 12 < 0, we can use the method of factoring.
Step 1: Rewrite the inequality in standard form:
x^2 - 8x + 12 < 0
Step 2: Factor the quadratic expression on the left-hand side of the inequality:
(x - 6)(x - 2) < 0
Step 3: Set each factor to zero and solve for x:
x - 6 = 0 or x - 2 = 0
x = 6 or x = 2
Step 4: Plot the critical points (x = 6 and x = 2) on a number line.
-∞ 2 6 +∞
──|───────────|─────|───>
Step 5: Test intervals between the critical points. Pick a value from each interval and substitute it into the inequality to determine the sign of the expression.
For x < 2: Let's choose x = 0
(0 - 6)(0 - 2) = (-6)(-2) = 12 > 0
Since the result is greater than 0, x < 2 is not part of the solution.
For 2 < x < 6: Let's choose x = 4
(4 - 6)(4 - 2) = (-2)(2) = -4 < 0
Since the result is less than 0, 2 < x < 6 is part of the solution.
For x > 6: Let's choose x = 7
(7 - 6)(7 - 2) = (1)(5) = 5 > 0
Since the result is greater than 0, x > 6 is not part of the solution.
Step 6: Determine the solution based on the sign of the expression on the intervals.
From the analysis of the intervals, we can see that the solution to the inequality is 2 < x < 6, expressed in interval notation as (2, 6). This means that x is greater than 2 and less than 6.
Therefore, the solution to the inequality x^2 - 8x + 12 < 0 is (2, 6).