12 multiple-choice questions, progressively harder.
Solve ∣2x−1∣>5|2x - 1| > 5∣2x−1∣>5.
Solution
Correct answer: A
A 'greater than' absolute value splits into two rays.
2x−1>5or2x−1<−52x - 1 > 5 \quad \text{or} \quad 2x - 1 < -52x−1>5or2x−1<−5
The first gives x>3x > 3x>3; the second gives x<−2x < -2x<−2.
Solve ∣x+1∣≤6|x + 1| \le 6∣x+1∣≤6.
Correct answer: D
A 'less than or equal to' absolute value gives one band.
−6≤x+1≤6-6 \le x + 1 \le 6−6≤x+1≤6
Subtracting 111 from every part gives −7≤x≤5-7 \le x \le 5−7≤x≤5.
What is the range of y=−2∣x+1∣+6y = -2|x + 1| + 6y=−2∣x+1∣+6?
The vertex is at (h,k)=(−1,6)(h, k) = (-1, 6)(h,k)=(−1,6), and a=−2<0a = -2 < 0a=−2<0 opens the V downward.
a<0 ⇒ y≤k=6a < 0 \;\Rightarrow\; y \le k = 6a<0⇒y≤k=6
Starting at the top point 666, the outputs only decrease.
The vertex of y=−4∣x+3∣−2y = -4|x + 3| - 2y=−4∣x+3∣−2 is at which point?
Match to y=a∣x−h∣+ky = a|x - h| + ky=a∣x−h∣+k. The inside x+3x + 3x+3 gives h=−3h = -3h=−3 and the constant is k=−2k = -2k=−2.
(h,k)=(−3,−2)(h, k) = (-3, -2)(h,k)=(−3,−2)
The factor −4-4−4 changes the width and opening but not the vertex.
The graph of y=a∣x∣y = a|x|y=a∣x∣ has its vertex at the origin and passes through (2,6)(2, 6)(2,6). What is aaa?
Substitute the point (2,6)(2, 6)(2,6) into y=a∣x∣y = a|x|y=a∣x∣.
6=a∣2∣=2a ⇒ a=36 = a|2| = 2a \;\Rightarrow\; a = 36=a∣2∣=2a⇒a=3
Solve 4∣x+1∣=204|x + 1| = 204∣x+1∣=20.
Correct answer: B
Isolate the bars by dividing by 444.
∣x+1∣=5 ⇒ x+1=5 or x+1=−5|x + 1| = 5 \;\Rightarrow\; x + 1 = 5 \text{ or } x + 1 = -5∣x+1∣=5⇒x+1=5 or x+1=−5
This gives x=4x = 4x=4 or x=−6x = -6x=−6.
Solve ∣x−1∣<0|x - 1| < 0∣x−1∣<0.
An absolute value is never negative, so it can never be strictly less than 000.
∣x−1∣≥0|x - 1| \ge 0∣x−1∣≥0
No value makes it negative, so there is no solution.
Solve ∣5x+10∣=0|5x + 10| = 0∣5x+10∣=0.
Correct answer: C
Only zero has absolute value zero.
5x+10=0 ⇒ 5x=−10 ⇒ x=−25x + 10 = 0 \;\Rightarrow\; 5x = -10 \;\Rightarrow\; x = -25x+10=0⇒5x=−10⇒x=−2
What is the range of y=∣x−3∣−7y = |x - 3| - 7y=∣x−3∣−7?
The vertex is at (3,−7)(3, -7)(3,−7), and a=1>0a = 1 > 0a=1>0 opens the V upward.
y≥k=−7y \ge k = -7y≥k=−7
Among these, which V opens the narrowest?
The larger ∣a∣|a|∣a∣ is, the steeper the branches and the narrower the V.
∣a∣=4 is the largest here|a| = 4 \text{ is the largest here}∣a∣=4 is the largest here
So y=4∣x∣y = 4|x|y=4∣x∣ opens the narrowest.
Solve ∣3x−9∣=6|3x - 9| = 6∣3x−9∣=6.
Split into two cases.
3x−9=6 ⇒ x=5,3x−9=−6 ⇒ 3x=3 ⇒ x=13x - 9 = 6 \;\Rightarrow\; x = 5, \qquad 3x - 9 = -6 \;\Rightarrow\; 3x = 3 \;\Rightarrow\; x = 13x−9=6⇒x=5,3x−9=−6⇒3x=3⇒x=1
The distance between two numbers aaa and bbb is ∣a−b∣|a - b|∣a−b∣. What is the distance between −3-3−3 and 555?
Substitute into ∣a−b∣|a - b|∣a−b∣ and simplify.
∣−3−5∣=∣−8∣=8|-3 - 5| = |-8| = 8∣−3−5∣=∣−8∣=8
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