Some equations involve absolute value. Consider the graph of the equation │a│= 4.
│a│= 4
The distance from 0 to a is 4 units. Therefore, a =
–4 or a = 4. When an equation has more than one solution, the solutions
are often written as a set {a, b}. The solution set is {–4,
4}.
Equations that involve absolute value can be solved by using
a number line or by writing them as compound sentences and solving them.
Example
1. Solve │x – 3│= 5. Check your
solution.
Alternative Solutions :
Method 1: Use a number line.
│x – 3│= 5 means the distance between x and 3 is 5 units.
So, to
find x on the number line, start at 3
and move 5 units in either direction.
Method 2: Write and solve a compound
sentence.
│x – 3│= 5 also means x – 3 = 5 or x – 3 = –5.
x – 3 = 5 or x – 3 = –5
x – 3 + 3 = 5 + 3 Add 3 to
each side. x – 3 + 3 = –5 + 3
x = 8 x
= – 2
Check:
Replace x with 8. Replace x with – 2.
│x – 3│= 5 │x – 3│= 5
│8 – 3│≟ 5 │2 – 3│≟ 5
│5│≟ 5 │5│≟ 5
5 = 5 ü 5 = 5 ü
The solution set is {2, 8}.
When solving an equation involving an absolute value symbol,
you can think of the absolute value bars as grouping symbols.
Example
2. Solve │a + 6│+ 5 = 12. Check your
solution.
Alternative Solutions :
To
solve the equation, first simplify the expression.
│a + 6│+ 5 = 12 Original equation
│a + 6│+ 5 – 5 = 12 – 5 Subtract 5 from each side.
│a + 6│= 7 Simplify.
Next,
write a compound sentence and solve it.
a + 6 = 7 or
a + 6 = –7
a + 6 – 6 = 7 – 6 Subtract
6 from each side. a + 6 – 6
= –7 – 6
a = 1 a = –13
Check: Replace a
with 1. Replace a with –13.
│a +
6│ + 5 = 12 │a + 6│+ 5 = 12
│1 + 6│ + 5 ≟ 12 │–13 + 6│+ 5 ≟ 12
│7│ + 5 ≟ 12 │–7│+ 5 ≟ 12
7 + 5 ≟ 12 7
+ 5 ≟ 12
12 = 12 ü 12 = 12 ü
The
solution set is {1, –13}.
Sometimes an equation has no solution. For example, │x│= - 6 is never true. The absolute value
of a number is always positive or zero. So, there is no replacement for x that
will make the sentence true. The solution set has no members. It is called the empty set, and
its symbols are { } or Ø.
Example
3. Solve │d│+ 7 = 2. Check your solution.
First,
simplify the expression.
│d│+ 7 = 2 Original equation
│d│+ 7 – 7
= 2 – 7 Subtract
7 from each side.
│d│1 = –5 Simplify.
This
sentence can never be true. The solution is the empty set or Ø.
Example
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4. Hydrogen can exist as a solid, liquid, or gas. For hydrogen to be a liquid,
its temperature must be within 2° of –257°C. Write and then solve an equation that
can be used to find the least and greatest temperatures at which hydrogen is a
liquid.
Alternative Solutions :
Let t
represent temperature. t differs from
–257 by exactly 2 degrees.
Write
an equation that represents the least and greatest temperatures.
t + 257 = 2 Rewrite the equation. t +
257 = –2
t + 257 – 257 = 2 – 257 Subtract
257. t + 257 – 257 = –2 – 257
t = –255 Simplify.
t = –259
The
solutions are –255 and –259. So, the least temperature for hydrogen to remain a
liquid is –259°C, and the greatest temperature is –255°C.
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