Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Let’s begin by solving the inequality x² - 5x + 6 < 0. Who can tell me how we should start?
We could factor the expression!
Exactly! When we factor, we rewrite it as (x - 2)(x - 3). Now, what should we do next?
We solve the equation to find the roots, right?
Yes! The roots are x = 2 and x = 3. Now we can identify the intervals. What are those?
They are x < 2, 2 < x < 3, and x > 3.
Correct! Next, choose test points in each interval to see where the inequality holds. What can we use for x < 2?
We can use x = 1.
Great choice! Now, when we plug that into the expression, what do we get?
It gives a positive result, so that part doesn't count for our solution.
Exactly! What about the interval 2 < x < 3?
We could try x = 2.5.
And what do we find?
That one gives a negative result, so it satisfies the inequality!
Right! Finally, for x > 3, what test point should we use?
We could use x = 4, and it gives a positive result.
Perfect! So what is our final answer?
2 < x < 3!
Let's summarize. We factored, found roots, analyzed intervals, and chose test points. Great work, everyone!
Moving on to our second example, we have 2x² - 8x + 6 ≥ 0. What are our initial steps?
Do we need to factor it or use the quadratic formula?
Yes! Let's use the quadratic formula. What is our quadratic formula?
x = [−b ± √(b² - 4ac)] / 2a.
Exactly! Here, a is 2, b is -8, and c is 6. Can someone calculate the roots?
We find x = 1 and x = 3!
Right! Now, we have our roots. Next, let’s find our intervals. What do we have?
We have x < 1, 1 < x < 3, and x > 3.
Great! Now, who would like to test the interval x < 1?
I can! We can use x = 0, and that gives a valid solution.
Excellent! And between 1 and 3, what will we use?
Let’s use x = 2. I think that will give a negative result.
Correct! And for x > 3?
Using x = 4 returns a valid result as well!
Exactly! So what is our final solution set?
x ≤ 1 or x ≥ 3!
Fantastic! Remember, we used the quadratic formula, tested intervals, and found our solution set. Well done, everyone!
Factor, find the roots, test the sign, to solve inequalities in record time!
Imagine two friends, 2 and 3, meeting at a party under the light of 'x'. They hold hands, making x stay between them and the party starts when they’re apart!
F-R-M: Factor, Root, Measure intervals - steps to solve quadratic inequalities!
S-I-R: Solve, Identify roots, Read intervals.
Term
What is a quadratic inequality?
Click to flip
Term
What are the roots of a quadratic expression?
Click to flip
This section provides detailed worked examples to illustrate the process of solving quadratic inequalities.
The section presents two comprehensive examples of solving quadratic inequalities. Each example is broken down step by step, demonstrating how to factor expressions, find roots, analyze sign changes, and represent solutions correctly. The goal is to solidify the understanding of solving such inequalities through practical application.
In this section, we explore worked examples that illustrate the process of solving quadratic inequalities, fundamental in mastering algebraic concepts involving quadratic expressions.
Factor the expression: We seek a factorization resulting in zero, thus we rewrite the expression as
(x - 2)(x - 3).
Solve the equation: Setting each factor to zero gives roots x = 2 and x = 3, dividing the number line into intervals:
Use sign analysis:
In conclusion, the solution set is 2 < x < 3.
Factor or use the quadratic formula: The roots are calculated using
x = (8 ± √(64 - 48)) / (4) = 1 or 3.
Test intervals:
Thus, the solution set is x ≤ 1 or x ≥ 3.