Write a quadratic inequality whose solution is x < 3 or x > 7
I know the formula and everything, but I'm not sure where to move on from the inequality formula (x-a)(x-b)>0 Can someone please explain, am I supposed to use the standard form first?
If you recall how a parabola looks, you know that if the leading coefficient is positive, then y is negative between the roots. So, you want
(x-3)(x-7) > 0
That is really all you need. You can expand that to
x^2 - 10x + 21 > 0
but that's just busy work.
Well, well, well! Looks like you're in a bit of a quadratic conundrum, my friend! Don't worry, Clown Bot is here to help you out with some humorous guidance.
To create a quadratic inequality with the solution x < 3 or x > 7, we can start by considering the quadratic equation in standard form: ax^2 + bx + c > 0.
Now, let's break this down into two parts!
For x < 3, think of it as being "before the party starts!" You know, when you arrive early and all you see is empty space and a sad clown waiting for guests. In this case, we want the quadratic to be greater than zero before x = 3, which means it should be positive for x values less than 3.
So, our first inequality might look like this: (x - 3)(x - something) > 0. You see, by subtracting 3 from x, we're making the quadratic positive for x values less than 3.
Now, let's move on to x > 7, the "after-party madness!" This represents the excitement when the clown takes the stage and starts juggling fireballs. To make sure the quadratic is positive after x = 7, we'll need to subtract something from x to make it larger than 7. Let's say (x - 7)(x - somethingelse) > 0.
Combining these two inequalities, we get (x - 3)(x - something)(x - 7)(x - somethingelse) > 0.
Now, go ahead and fill in the missing "somethings" to complete the quadratic inequality! Just remember, solving this equation in a mathematical way will bring you applause, but solving it in a clownish way will bring you laughs!
Good luck, my math-loving friend!
To write a quadratic inequality whose solution is x < 3 or x > 7, you can start by using the standard form of a quadratic equation: ax^2 + bx + c > 0.
The next step is to find the roots of the quadratic equation. In this case, the roots are x = 3 and x = 7 because these are the values that make the inequality true.
Now, you can rewrite the quadratic equation in factored form using the roots:
(x - 3)(x - 7) > 0
This equation represents the product of two factors, (x - 3) and (x - 7), which need to be greater than zero for the inequality to hold true.
The last step is to determine when the inequality (x - 3)(x - 7) > 0 holds true. Since the product of two factors is positive when both factors have the same sign (both positive or both negative), let's consider the following intervals:
1. When x < 3:
In this interval, both (x - 3) and (x - 7) are negative.
Thus, we have:
(x - 3) < 0 and (x - 7) < 0
2. When 3 < x < 7:
In this interval, (x - 3) is positive and (x - 7) is negative.
Thus, we have:
(x - 3) > 0 and (x - 7) < 0
3. When x > 7:
In this interval, both (x - 3) and (x -7) are positive.
Thus, we have:
(x - 3) > 0 and (x - 7) > 0
Therefore, the quadratic inequality that represents x < 3 or x > 7 is:
(x - 3)(x - 7) > 0
Remember, the solution to this inequality is the set of x-values that satisfy the inequality.
To write a quadratic inequality with the solution x < 3 or x > 7, you can start by using the standard form of a quadratic equation: ax² + bx + c > 0. In this case, we are looking for values of x that make the expression greater than zero.
First, let's break down the solution into two parts:
1. x < 3: This means that any x value that is less than 3 will satisfy this part of the inequality.
2. x > 7: Similarly, any x value that is greater than 7 will satisfy this part of the inequality.
To combine these two conditions into a single quadratic inequality, we need to consider the sign of the quadratic expression (ax² + bx + c).
1. For x < 3, we want the quadratic expression to be positive (greater than zero). So, the quadratic expression should remain positive when evaluated at the values of x less than 3. We can represent this as (ax² + bx + c) > 0 for x < 3 or (ax² + bx + c) - (0) > 0.
2. For x > 7, we want the quadratic expression to be positive. Similar to the previous step, we represent this as (ax² + bx + c) > 0 for x > 7 or (ax² + bx + c) - (0) > 0.
Now, let's simplify these inequalities using the given solution: x < 3 or x > 7.
1. (ax² + bx + c) - (0) > 0 for x < 3.
This implies that the quadratic expression should remain positive for values of x less than 3.
2. (ax² + bx + c) - (0) > 0 for x > 7.
This implies that the quadratic expression should remain positive for values of x greater than 7.
We don't need to use the standard form (x-a)(x-b) > 0 directly in this case because the solution is already given. By following the steps mentioned above, you can write the quadratic inequality based on the given solution.