Apply the definition of an absolute value, then solve each equation for x.
B
b
Apply the definition of an absolute value, then solve each equation for x.
C
c
Apply the definition of an absolute value, then solve each equation for x.
D
d
Can an absolute value be equal to a negative number?
A
a
x=-2, 8
B
b
x=± 7
C
c
x=-3, 1
D
d
No real solution
Practice makes perfect
a
To solve the equation, we need to apply the definition of absolute value.
|x| = a ⇒ x=a or x= - a
By applying it to the equation |x-3|=5 we will get two equations to solve.
|x-3|=5
↙ ↘
x-3= 5 x-3=- 5
Let's begin by solving the left-hand side equation.
Let's check our solutions by plugging them into the original equation. If we get a true statement, it will imply that the solution is correct. Otherwise, it will be incorrect. Let's begin with x=-2.
Again, we will use the definition of an absolute value.
|x| = a ⇒ x=a or x= - aThis time, before applying the definition, we can divide both sides of the equation by 5. That way, we isolate |x| on the left side.
5|x|=35 ⇒ |x| = 7
Then, when we apply the definition of absolute value, we instantly get the solutions.
|x|=7
↙ ↘
x= 7 x=- 7
Checking Our Answer
Checking our Solutions
As above, we will check that our solutions are correct by replacing them into the original equation. Let's begin with x=- 7.
As before, we need to apply the definition of an absolute value.
|x| = a ⇒ x=a or x= - a
As in part A, by applying it to the equation |x+1|=2 we will get two equations to solve.
|x+1|=2
↙ ↘
x+1= 2 x+1 = - 2
Let's begin by solving the left-hand side equation.
Let's check our solutions by plugging them into the original equation. If we get a true statement, it will imply that the solution is correct. Otherwise, it will be incorrect. Let's begin with x=-3.
This time, notice that the right-hand side of the equation is a negative number. Since the absolute value of a number is always non-negative, it is impossible to solve this equation.
ccc
Non-negative & & Negative
|x+3| & ≠& - 2
Therefore, the equation |x+3|=- 2 has no real solution.
