Question 1 :
If |1 - x| > 4 and x is positive, what is one possible value of x?
Answer :
|1 - x| > 4
1 - x > 4 or 1 - x < -4
1 - x > 4 (1 - x) - 1 > 4 - 1 1 - x - 1 > 4 - 1 -x > 3 x < -3 |
1 - x < -4 (1 - x) - 1 < -4 - 1 1 - x - 1 < -5 -x < -5 x > 5 |
Since x is positive, one possible value of x satisfying x > 5 is 6. That is
x = 6
Question 2 :
0 < |5/x| < 1
Which of the following values of x satisfies the inequality above?
A) 2
B) 3
C) -3
D) -8
Answer :
0 < |5/x| < 1
A) 2 :
Substitute x = 2 in |5/x|.
|5/2| = 5/2 > 1
When x = 2, the value of |5/x| is not between 0 and 1.
2 does not work.
B) 3 :
Substitute x = 3 in |5/x|.
|5/3| = 5/3 > 1
When x = 3, the value of |5/x| is not between 0 and 1.
3 does not work.
C) -3 :
Substitute x = -3 in |5/x|.
|5/(-3)| = |-5/3|
= 5/3 > 1
When x = -3, the value of |5/x| is not between 0 and 1.
-3 does not work.
D) -8 :
Substitute x = 8 in |5/x|.
|5/8| = |5/8|
= 5/8
The value 5/8 is between 0 and 1, that is
0 < 5/8 < 1
When x = -8, the value of |5/x| is between 0 and 1.
-8 works.
The correct answer choice is (D).
Question 3 :
A bakery standardizes muffins to weigh between 1¾ and 2¼ounces. If m is the weight of muffin from this bakery, which of the following inequalities expresses the following possible values of m?
A) |m - 1¾| < ¼
B) | m - 2| < ¼
C) |m - 2| < ½
D) |m - 1¾| < ½
Answer :
It's kind of solving absolute value inequality in reverse order. Here, the boundary values of m are given and we have to find the absolute value inequality which gives the given boundary values ¾ and 2¼.
Find the middle value of the two boundary values.
Middle value of 1¾ and 2¼ is their average :
= (1¾ + 2¼) ÷ 2
= 4 ÷ 2
= 2
Since 1¾ and 2¼ are the boundary values of m,
1¾ < m < 2¼
Subtract the middle value.
(1¾ - 2) < (m - 2) < (2¼ - 2)
-¼ < (m - 2) < ¼
We know that
-A < x < A ----> |x| < A
So, we have
-¼ < (m - 2) < ¼ ----> |m - 2| < ¼
The correct answer choice is (B).
Question 4 :
If |x + 3| < 2, which of the following could be the value of |x|?
A) 1
B) 4
C) 6
D) 10
Answer :
|x + 3| < 2
x + 3 < 2 or x + 3 > -2
x < -1 or x > -5
Combined inequality of x < -1 and x > -5 :
-5 < x < -1
The possible integer values of x are -4, -3, -2.
Substitute x = -4, -3, -2, -1 in |x|.
|-4| = 4
|-3| = 3
|-2| = 2
The correct answer choice is (B).
Question 5 :
Which of the following could be the equation of the function graphed in the xy-plane above?
A) y = -|x - 1| - 2
B) y = |x - 1| - 2
C) y = |x - 1| + 2
D) y = |x + 1| - 2
Answer :
The graph of y = |x| is a V-shape centered at the origin.
The graph above is also V-shaped. But, it is shifted 1 unit to the right and 2 units down.
After shifting 1 unit to the right, y = |x| becomes
y = |x - 1|
Further shifting of 2 units down,
y = |x - 1| - 2
The correct answer choice is (B).
Question 6 :
If |x - 10| = y, x < 10, then which of the following is equivalent to (y - x)?
A) -10
B) 10
C) 2y - 10
D) 10 - 2y
Answer :
Assume some value for x such that x < 10, say x = 3.
In |x - 10| = y, substitute x = 3 and find the value of y.
y = |3 - 10|
y = |-7|
y = 7
Find the value of (y - x).
y - x = 7 - 3
y - x = 4
Among the given answer choices, pick the one which gives the result 4 when y = 7.
The correct answer choice is (C).
Note :
Instead of assuming x = 3, you can assume any value for x such that x < 10 and solve this problem. You will get the same answer.
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