please help
Let A = [0,3), B = { x ∈ ℝ : -11 < 2x +1 ≤ 11 } and C be the set of numbers for which x^2 - x - 20 < 0 .
I need to enter the following intervals in standard notation, using round brackets, square brackets or one of each.
B ∪ (A ∩ C)
(B ∪ A) ∩ C
Thankyou!
Let A = [0,3), B = { x ∈ ℝ : -11 < 2x +1 ≤ 11 } and C be the set of numbers for which x^2 - x - 20 < 0 .
I need to enter the following intervals in standard notation, using round brackets, square brackets or one of each.
B ∪ (A ∩ C)
(B ∪ A) ∩ C
Thankyou!
-
First get B and C in the same form as A.
Add -1 to each of the three expressions in
-11 < 2x +1 ≤ 11
-12 < 2x ≤ 10
-6 < x ≤ 5
Thus B = (-6, 5]
Now x² - x - 20 = (x + 4)(x - 5) which is negative when x is between -4 and 5.
i.e. C = (-4, 5)
Sketching these sets on the number axis would help us see that
A is contained in C, and both of them are contained in B.
Hence
A ∩ C = A
and
and so
B ∪ (A ∩ C) = B ∪ A = B = (-6, 5)
For the second expression,
B ∪ A = B
hence
(B ∪ A) ∩ C = B ∩ C = C = (-4, 5)
Add -1 to each of the three expressions in
-11 < 2x +1 ≤ 11
-12 < 2x ≤ 10
-6 < x ≤ 5
Thus B = (-6, 5]
Now x² - x - 20 = (x + 4)(x - 5) which is negative when x is between -4 and 5.
i.e. C = (-4, 5)
Sketching these sets on the number axis would help us see that
A is contained in C, and both of them are contained in B.
Hence
A ∩ C = A
and
and so
B ∪ (A ∩ C) = B ∪ A = B = (-6, 5)
For the second expression,
B ∪ A = B
hence
(B ∪ A) ∩ C = B ∩ C = C = (-4, 5)