#1. solve x^2-1<0 I got (-1,1)
#2 x(x+1) >20
I got (-infinity, -4) U (5,infinity)
#3
what is the domain of this function? f(x)= squre root of (x-3x^2)
i got [-1/3,0]
thanks!!!
#1
correct
for #2
x^2 + x - 20 > 0
(x+5)(x-4) > 0
critical values are -5 and +4
In my notation it would be
x < -5 OR x > 4
Your (-infinity, -4) U (5,infinity) should then be
(-infinity, -5) U (4,infinity)
#3
correct
Thanks so much!
To solve the inequality x^2 - 1 < 0, you need to find the values of x that make the expression less than zero. Here's how you can do it:
Step 1: Set the expression x^2 - 1 to be equal to zero and solve for x:
x^2 - 1 = 0
(x + 1)(x - 1) = 0
Step 2: Find the critical points by setting each factor equal to zero:
x + 1 = 0 => x = -1
x - 1 = 0 => x = 1
These critical points divide the number line into three intervals: (-infinity, -1), (-1, 1), and (1, infinity).
Step 3: Test a value within each interval to determine if the expression is positive or negative. For example, let's test x = 0:
For x < -1: Test x = -2:
(-2)^2 - 1 = 4 - 1 = 3 > 0
For -1 < x < 1: Test x = 0:
(0)^2 - 1 = -1 < 0
For x > 1: Test x = 2:
(2)^2 - 1 = 4 - 1 = 3 > 0
From the test results, we can see that the expression is positive in the interval (-infinity, -1) and (1, infinity), and negative in the interval (-1, 1). Therefore, the solution to x^2 - 1 < 0 is the interval (-1, 1).
Now, let's move on to the second question: x(x+1) > 20. Here's how you can solve it:
Step 1: Move all the terms to one side of the inequality:
x(x + 1) - 20 > 0
Step 2: Expand and simplify the expression:
x^2 + x - 20 > 0
Step 3: Factor the quadratic expression:
(x + 5)(x - 4) > 0
Step 4: Find the critical points by setting each factor equal to zero:
x + 5 = 0 => x = -5
x - 4 = 0 => x = 4
These critical points divide the number line into three intervals: (-infinity, -5), (-5, 4), and (4, infinity).
Step 5: Test a value within each interval to determine if the expression is positive or negative. For example, let's test x = 0:
For x < -5: Test x = -6:
(-6 + 5)(-6 - 4) = -1 * -10 = 10 > 0
For -5 < x < 4: Test x = 0:
(0 + 5)(0 - 4) = 5 * -4 = -20 < 0
For x > 4: Test x = 5:
(5 + 5)(5 - 4) = 10 * 1 = 10 > 0
From the test results, we can see that the expression is positive in the intervals (-infinity, -5) and (4, infinity), and negative in the interval (-5, 4). Therefore, the solution to x(x + 1) > 20 is the union of the intervals (-infinity, -5) and (4, infinity), which is (-infinity, -5) U (4, infinity).
Finally, let's find the domain for the function f(x) = √(x - 3x^2). The domain refers to the set of all possible values of x for which the function is defined. In this case, we need to find the values of x that do not result in any undefined operations, such as taking the square root of a negative number or dividing by zero.
Step 1: Determine the values of x that would make the expression inside the square root zero or negative:
x - 3x^2 ≤ 0
Step 2: Factor the expression:
x(1 - 3x) ≤ 0
Step 3: Find the critical points by setting each factor equal to zero:
x = 0
1 - 3x = 0 => x = 1/3
These are the critical points that divide the number line into three intervals: (-infinity, 0), (0, 1/3), and (1/3, infinity).
Step 4: Test a value within each interval to determine if the expression is positive or non-positive. For example, let's test x = 1, which is in the last interval:
For x < 0: Test x = -1:
√(-1 - 3(-1)^2) = √(-1 - 3) = √(-4) (undefined)
For 0 < x < 1/3: Test x = 1/4:
√(1/4 - 3(1/4)^2) = √(1/4 - 3/16) = √(1/4 - 3/16) = √(4/16 - 3/16) = √(1/16) = 1/4 > 0
For x > 1/3: Test x = 1:
√(1 - 3(1)^2) = √(1 - 3) = √(-2) (undefined)
From the test results, we can see that the expression is non-positive in the interval (0, 1/3) and undefined in the intervals (-infinity, 0) and (1/3, infinity). Therefore, the domain of the function f(x) = √(x - 3x^2) is [-infinity, 0] U (0, 1/3].
I hope this explanation helps!