I just cannot seem to be able to grasp the concept of solving polynomial inequalities. Can someone please explain, step by step, how to solve them? Here's a problem I can't solve. Please use this as an example:
(x-2)(x-5)<0
I cannot thank you enough for helping me with this. I have a test on this tomorrow and this is the only section I don't get. Thank you for all the help!
First of all, sketch a graph of
y = (x-2)(x-5)
This will be a parabola because it is a quadratic, a x^2 + b x + c
for large negative x, y is big positive
it is zero at x = 2
then it is negative for a while
it is then zero at x = 5
then it goes big positive
so
negative between x = 2 and x = 5
Thank you so much, Damon! You really helped me out! : )
I used to teach it with a slightly different approach because it worked well with multiple factors.
for example I will add another factor to your problem
(x-2)(x-5)(x+2) > 0
the x-intercepts of the corresponding function would be x = -2,2, and 5
mark those on the x-axis
there are now 4 sections to my x-axis
1. all values below -2
2. all values between -2 and 2
3. all values between 2 and 5
4. all values greater than 5
now pick an arbitrary value in each section and test it. You don't have to actually evaluate it, just worry about + or - (matching the > or <)
1. let x=-5 ---- (-)(-)(-) = - (does not work)
2. let x = 0 ---(-)(-)(+) = + works
3. let x = 3 ---(+)(-)(+) = - no
4. let x = 6 --- (+)(+)(+) = + yes
so -2 < x < 2 OR x > 5
for quadratic inequalities, I would also use the method outlined by Damon, but for cubics and higher, the above method works very nicely.
Of course if the function does not factor it would be a mess anyway.
Thank you, Reiny. My teacher taught me something very similar to your method and now I understand it. Thank you for your help! : )
Sure! I'm here to help. Solving polynomial inequalities involves several steps. Let's solve the given example step by step:
Step 1: Factor the polynomial.
In this case, we have (x-2)(x-5)<0. This polynomial is already factored, so we can move to the next step.
Step 2: Determine the critical points.
To solve the inequality, we need to find the critical points, which are the values of x that make the inequality equal to zero. In this case, we have two critical points: x = 2 and x = 5.
Step 3: Create intervals.
Next, we will create intervals on the number line based on the critical points. We have three intervals: (-∞, 2), (2, 5), and (5, ∞).
Step 4: Test each interval.
Now, we will test a value within each interval to determine the sign of the expression (x-2)(x-5). Let's choose a value less than 2, like x = 0, a value between 2 and 5, like x = 3, and a value greater than 5, like x = 6.
Step 5: Evaluate the expression within each interval.
For the interval (-∞, 2), substitute x = 0 into the expression (x-2)(x-5).
(0-2)(0-5) = (-2)(-5) = 10, which is positive.
So, the expression is positive in the interval (-∞, 2).
For the interval (2, 5), substitute x = 3 into the expression (x-2)(x-5).
(3-2)(3-5) = (1)(-2) = -2, which is negative.
So, the expression is negative in the interval (2, 5).
For the interval (5, ∞), substitute x = 6 into the expression (x-2)(x-5).
(6-2)(6-5) = (4)(1) = 4, which is positive.
So, the expression is positive in the interval (5, ∞).
Step 6: Determine the solution.
Based on the signs obtained in Step 5, we can determine the solution to the inequality. The solution is the set of intervals where the expression is less than zero, since we have (x-2)(x-5)<0.
Therefore, the solution is (2, 5).
In summary, the solution to the inequality (x-2)(x-5)<0 is x ∈ (2, 5). This means that x is any value between 2 and 5, excluding 2 and 5 themselves.
I hope this step-by-step explanation helps you understand how to solve polynomial inequalities. Good luck with your test!