Write a quadratic inequality whose solution is x < 3 or x > 7.
(x-3)(x-7) > 0
see your previous post for why.
To write a quadratic inequality with the given solution, we can break it down into two inequalities:
1. x < 3
2. x > 7
Let's start with the first inequality:
1. x < 3
Since this is a quadratic inequality, we can set it up in standard form ax^2 + bx + c < 0. To do this, we rewrite the inequality as:
x - 3 < 0
Next, let's work on the second inequality:
2. x > 7
Again, we can set it up in standard form ax^2 + bx + c > 0:
x - 7 > 0
Combining these two inequalities, we can write the quadratic inequality as:
(x - 3)(x - 7) < 0
This is the quadratic inequality whose solution is x < 3 or x > 7.
To write a quadratic inequality with the solution x < 3 or x > 7, we need to consider that the quadratic function opens upward or downward depending on its leading coefficient.
Let's consider the case where the quadratic opens upward. In this case, we want the graph of the quadratic to be below the x-axis (negative values of y) between x = 3 and x = 7, and above the x-axis (positive values of y) elsewhere.
To create such an inequality, we can use (x - a)(x - b) < 0, where a = 3 and b = 7.
(x - 3)(x - 7) < 0
Now, we can solve this inequality to find the values of x that satisfy it. This can be done by finding the critical points, which are the x-values where the quadratic expression equals zero (x - 3)(x - 7) = 0. In this case, the critical points are x = 3 and x = 7.
Next, we create a number line and test the inequality at various intervals.
Using a test value less than 3, let's say x = 2:
(2 - 3)(2 - 7) < 0
(-1)(-5) < 0
5 < 0
This test fails, so we can exclude the interval x < 3.
Using a test value between 3 and 7, let's say x = 5:
(5 - 3)(5 - 7) < 0
(2)(-2) < 0
-4 < 0
This test succeeds, so the solution includes the interval 3 < x < 7.
Using a test value greater than 7, let's say x = 8:
(8 - 3)(8 - 7) < 0
(5)(1) < 0
5 < 0
This test fails, so we can exclude the interval x > 7.
Putting it all together, the quadratic inequality whose solution is x < 3 or x > 7 is:
(x - 3)(x - 7) < 0, where 3 < x < 7.