Absolute Value Equations Practice

What are the solutions of the following equations?

1. $2|4x+5|=0$

   Choose 1 answer:

   1. $x=-\dfrac{4}{5}$ or $x=\dfrac{5}{4}$
   2. Only $x=-\dfrac{5}{4}$
   3. Only $x=-\dfrac{4}{5}$

   4. There are no solutions.

2. $8 = 3|x-7| +4$

   Choose 1 answer:

   1. $x=-\dfrac{25}{3}$ or $x=\dfrac{25}{3}$
   2. $x=\dfrac{17}{3}$ or $x=\dfrac{25}{3}$
      
   3. Only $x=\dfrac{25}{3}$

   4. There are no solutions.

3. $|3x-2|-9=-10$

   Choose 1 answer:

   1. $x=\dfrac{1}{3}$ or $x=1$
   2. $x=\dfrac{1}{3}$ or $x=-\dfrac{1}{3}$
   3. Only $x=\dfrac{1}{3}$

   4. There are no solutions.

4. $6+4|x+1|=6$

   Choose 1 answer:

   1. $x=-\dfrac{5}{2}$ or $x=\dfrac{1}{2}$
   2. $x=-1$ or $x=\dfrac{1}{2}$
      
   3. Only $x=-1$
   4. There are no solutions.

Source: Khan Academy, https://www.khanacademy.org/math/college-algebra/xa5dd2923c88e7aa8:linear-equations-and-inequalities/xa5dd2923c88e7aa8:absolute-value-equations/e/absolute-value-equations
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1. 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 $x=-\dfrac{5}{4}$.

   ---

2. 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 $x=\dfrac{25}{3}$.

   ---

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.

   ---

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

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

   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 $x=-1$.