Break down the given absolute value equation into two separate equations.
x=5
Practice makes perfect
When solving an equation involving absolute value expressions, we should consider what would happen if we removed the absolute value symbols. Let's look at an example equation.
|ax+b|=|cx+d|
Although we can make 4 statements about this equation, there are actually only two possible cases to consider.
Statement
Result
Both absolute values are positive.
ax+b=cx+d
Both absolute values are negative.
-(ax+b)=-(cx+d)
Only the left-hand side is negative.
-(ax+b)=cx+d
Only the right-hand side is negative.
ax+b=-(cx+d)
Because of the Properties of Equality, when the absolute values of two expressions are equal, either the expressions are equal or the opposites of the expressions are equal. Now let's consider these two cases for the given equation.
Given Equation:&& |x-6|&=|- x+4|
First Equation:&& x-6&= - x+4
Second Equation:&& x-6&=-(- x+4)
We will solve each of these equations by graphing separately.
First Equation
To graph the first equation, we will create two functions out of the left- and right-hand sides of the equation.
y=x-6 and y=- x+4
The x-coordinate where the graphs of these functions intersect is the solution to our equation.
From the graph, we can see that the graphs intersect at (5,- 1) which means the solution to the equation is x=5. Let's check whether it is correct by substituting it into the original equation.
In order to graph the second equation, we will again create functions out of the left- and right-hand sides of the equation.
y=x-6 and y=- (- x+4)
Once more, the x-coordinate where the graphs of these functions intersect is the solution to our equation. Since one of the given equations is not in slope-intercept form, let's rewrite it so that it will be easier to identify its slope and y-intercept.
We have found that these lines are parallel, and therefore will never have a point of intersection. This means that for this possible configuration of the original equation, there is no solution. Therefore, x=5 is the only solution to the given absolute value equation.
