Absolute Value Equations Practice

The strategy

• First, we manipulate the equation to isolate the absolute value term. To put it simply, we rewrite the equation in the general form $|x-a|=b$.
• If $b\geq0$, we continue by solving the equations ‍$x-a=b$ and ‍$x-a=-b$.
• Instead, if $b$, there are no solutions because the absolute value can never be negative.

Isolating the absolute value term

$\begin{aligned}2{|4x+5|}&=0\\\\{|4x+5|}&=0\end{aligned}$

Since the absolute value term equals 0, we can continue solving the equation.

Solving for $x$

Since 0 is its own additive inverse (which is another way of saying that ‍0 is the same as ‍-0)), the absolute value equation above corresponds to only one linear equation, which is $4x+5=0$.

Therefore, the only solution to the absolute value equation is $x=-\dfrac{5}{4}$.

Conclusion

The only solution to the given equation is $x=-\dfrac{5}{4}$.

The strategy

• First, we manipulate the equation to isolate the absolute value term. To put it simply, we rewrite the equation in the general form $|x-a|=b$.
• If $b\geq0$, we continue by solving the equations ‍$x-a=b$ and ‍$x-a=-b$.
• Instead, if $b$, there are no solutions because the absolute value can never be negative.

Isolating the absolute value term

$\begin{aligned}8 &= 3{|x-7|} +4\\\\4 &= 3{|x-7|}\\\\\dfrac{4}{3} &= {|x-7|}\end{aligned}$

Since the absolute value term equals a positive number, we can continue solving the equation.

Solving for $x$

The absolute value equation now turns into two linear equations.

• The first equation is $\dfrac{4}{3}=x-7$ and its solution is $x=\dfrac{25}{3}$.
• The second equation is $-\dfrac{4}{3}=x-7$ and its solution is $x=\dfrac{17}{3}$.

Conclusion

The only solution to the given equation is $x=\dfrac{17}{3}$ or $x=\dfrac{25}{3}$.

The strategy

• First, we manipulate the equation to isolate the absolute value term. To put it simply, we rewrite the equation in the general form $|x-a|=b$.
• If $b\geq0$, we continue by solving the equations ‍$x-a=b$ and ‍$x-a=-b$.
• Instead, if $b$, there are no solutions because the absolute value can never be negative.

Isolating the absolute value term

$\begin{aligned}{|3x-2|}-9&=-10\\\\{|3x-2|}&=-1\end{aligned}$

Since the absolute value term equals a negative number, the equation has no solutions.

Conclusion

There are no solutions to the given equation.

The strategy

• First, we manipulate the equation to isolate the absolute value term. To put it simply, we rewrite the equation in the general form $|x-a|=b$.
• If $b\geq0$, we continue by solving the equations ‍$x-a=b$ and ‍$x-a=-b$.
• Instead, if $b$, there are no solutions because the absolute value can never be negative.

Isolating the absolute value term

$\begin{aligned}6+4{|x+1|}&=6\\\\4{|x+1|}&=0\\\\{|x+1|}&=0\end{aligned}$

Since the absolute value term equals 0, we can continue solving the equation.

Solving for $x$

Since 0 is its own additive inverse (which is another way of saying that ‍0 is the same as ‍-0)), the absolute value equation above corresponds to only one linear equation, which is $x+1=0$.

Therefore, the only solution to the absolute value equation is $x=-1$.

Conclusion

The only solution to the given equation is $x=-1$.