12 multiple-choice questions, progressively harder.
Solve ∣3x−5∣=7|3x - 5| = 7∣3x−5∣=7.
Solution
Correct answer: B
Split the positive right side into two cases.
3x−5=7 ⇒ x=4,3x−5=−7 ⇒ 3x=−2 ⇒ x=−233x - 5 = 7 \;\Rightarrow\; x = 4, \qquad 3x - 5 = -7 \;\Rightarrow\; 3x = -2 \;\Rightarrow\; x = -\tfrac{2}{3}3x−5=7⇒x=4,3x−5=−7⇒3x=−2⇒x=−32
How many solutions does 2∣x−1∣+5=52|x - 1| + 5 = 52∣x−1∣+5=5 have?
Correct answer: C
Isolate the bars.
2∣x−1∣=0 ⇒ ∣x−1∣=0 ⇒ x=12|x - 1| = 0 \;\Rightarrow\; |x - 1| = 0 \;\Rightarrow\; x = 12∣x−1∣=0⇒∣x−1∣=0⇒x=1
A right side of 000 gives exactly one solution.
Solve ∣7−2x∣=3|7 - 2x| = 3∣7−2x∣=3.
Split into two cases.
7−2x=3 ⇒ x=2,7−2x=−3 ⇒ −2x=−10 ⇒ x=57 - 2x = 3 \;\Rightarrow\; x = 2, \qquad 7 - 2x = -3 \;\Rightarrow\; -2x = -10 \;\Rightarrow\; x = 57−2x=3⇒x=2,7−2x=−3⇒−2x=−10⇒x=5
What is the range of y=3∣x−2∣−4y = 3|x - 2| - 4y=3∣x−2∣−4?
Correct answer: D
The vertex is at (2,−4)(2, -4)(2,−4), and a=3>0a = 3 > 0a=3>0 opens the V upward.
y≥k=−4y \ge k = -4y≥k=−4
Solve ∣x∣>6|x| > 6∣x∣>6.
Correct answer: A
A 'greater than' absolute value gives two rays.
x>6orx<−6x > 6 \quad \text{or} \quad x < -6x>6orx<−6
These are the numbers more than six units from zero.
Solve ∣2x+3∣=−5|2x + 3| = -5∣2x+3∣=−5.
An absolute value is never negative, so it cannot equal −5-5−5.
∣2x+3∣≥0>−5|2x + 3| \ge 0 > -5∣2x+3∣≥0>−5
The negative right side means there is no solution.
Where does the graph of y=∣x+2∣−3y = |x + 2| - 3y=∣x+2∣−3 cross the x-axis?
Set y=0y = 0y=0 and solve the absolute value equation.
∣x+2∣−3=0 ⇒ ∣x+2∣=3 ⇒ x+2=3 or x+2=−3|x + 2| - 3 = 0 \;\Rightarrow\; |x + 2| = 3 \;\Rightarrow\; x + 2 = 3 \text{ or } x + 2 = -3∣x+2∣−3=0⇒∣x+2∣=3⇒x+2=3 or x+2=−3
This gives x=1x = 1x=1 or x=−5x = -5x=−5.
Solve ∣6−x∣=2|6 - x| = 2∣6−x∣=2.
6−x=2 ⇒ x=4,6−x=−2 ⇒ x=86 - x = 2 \;\Rightarrow\; x = 4, \qquad 6 - x = -2 \;\Rightarrow\; x = 86−x=2⇒x=4,6−x=−2⇒x=8
Solve ∣x−9∣=0|x - 9| = 0∣x−9∣=0.
Only zero has absolute value zero.
x−9=0 ⇒ x=9x - 9 = 0 \;\Rightarrow\; x = 9x−9=0⇒x=9
Solve ∣2x−1∣≥7|2x - 1| \ge 7∣2x−1∣≥7.
A 'greater than or equal to' absolute value gives two rays.
2x−1≥7or2x−1≤−72x - 1 \ge 7 \quad \text{or} \quad 2x - 1 \le -72x−1≥7or2x−1≤−7
The first gives x≥4x \ge 4x≥4; the second gives 2x≤−62x \le -62x≤−6, so x≤−3x \le -3x≤−3.
For which value of ccc does ∣2x−4∣=c|2x - 4| = c∣2x−4∣=c have no solution?
An absolute value equation has no solution exactly when the right side is negative.
c<0 ⇒ no solutionc < 0 \;\Rightarrow\; \text{no solution}c<0⇒no solution
Among the choices, only c=−3c = -3c=−3 is negative; c=0c = 0c=0 gives one solution and the positive values give two.
Solve ∣x+4∣=2x−1|x + 4| = 2x - 1∣x+4∣=2x−1.
Split into two cases, then check that the right side 2x−12x - 12x−1 is not negative.
x+4=2x−1 ⇒ x=5,x+4=−(2x−1) ⇒ x=−1x + 4 = 2x - 1 \;\Rightarrow\; x = 5, \qquad x + 4 = -(2x - 1) \;\Rightarrow\; x = -1x+4=2x−1⇒x=5,x+4=−(2x−1)⇒x=−1
Testing, x=5x = 5x=5 gives ∣9∣=9=2(5)−1|9| = 9 = 2(5) - 1∣9∣=9=2(5)−1, but x=−1x = -1x=−1 makes the right side negative, so it fails. Only x=5x = 5x=5 works.
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