➕ Algebra 1 · Inequalities
Absolute Value Equations flashcards
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All 15 flashcards at a glance
Card 1
What does |x| mean?
The distance from x to 0 on the number line. It is never negative.
Card 2
Evaluate |−13| and −|13|.
|−13| = 13, because −13 is 13 units from 0. −|13| = −13, because the bars are worked out first and then the minus sign applies.
Card 3
Does |x + 1| = −9 have a solution?
No. An absolute value is a distance, and a distance is never negative.
Card 4
How many solutions does |x − 9| = 0 have?
One: x = 9. Only 9 is 0 units from 9.
Card 5
Solve |x + 5| = 8.
x = 3 or x = −13. Write x + 5 = 8 or x + 5 = −8.
Card 6
Solve |2x − 3| = 11.
x = 7 or x = −4. From 2x − 3 = 11 you get x = 7. From 2x − 3 = −11 you get 2x = −8, so x = −4.
Card 7
Solve 5|x| − 3 = 22.
x = 5 or x = −5. Add 3: 5|x| = 25. Divide by 5: |x| = 5. Then split.
Card 8
Solve 4|x − 6| + 2 = 14.
x = 9 or x = 3. Subtract 2: 4|x − 6| = 12. Divide by 4: |x − 6| = 3. Then x − 6 = 3 or x − 6 = −3.
Card 9
Why can’t you split 3|x − 2| = 21 into x − 2 = 21 or x − 2 = −21?
The bars are not alone. Divide by 3 first: |x − 2| = 7. Then x − 2 = 7 or x − 2 = −7, so x = 9 or x = −5. The wrong split gives 23 and −19, and 3|23 − 2| = 63, not 21.
Card 10
On a number line, what do the solutions of |x − 4| = 2 look like?
The two points that are 2 units from 4: x = 2 and x = 6.
Card 11
A pipe must be 40 centimeters long, give or take 0.5 centimeter. What equation gives the two limiting lengths L, and what are they?
|L − 40| = 0.5, so L = 39.5 or L = 40.5 centimeters. Every length between them also passes: 39.5 ≤ L ≤ 40.5.
Card 12
How do you check a solution of an absolute value equation?
Substitute it into the original equation and work out each side: first the arithmetic inside the bars, then the absolute value, then the rest.
Card 13
Why does |x| = a have no solution when a < 0?
|x| is a distance from 0, and a distance is never negative.
Card 14
Solve 10 − 2|x + 3| = 4.
x = 0 or x = −6. Subtract 10: −2|x + 3| = −6. Divide by −2: |x + 3| = 3. Then x + 3 = 3 or x + 3 = −3.
Card 15
In this lesson’s BARS memory trick, what do the letters stand for?
Bars alone, Ask about the number on the other side (positive, zero or negative), Rewrite as two equations (when it is positive), Solve both and check both.