Solving Absolute Value Equations

Solving Absolute Value Equations exercise

• What is the temperature at the vacation resort?

  Hints

  Use a number line to model absolute value.

  Check your work. The temperature difference from $60^{\circ}$F has to be $30^{\circ}$F.

  If you get stuck, refer back to the definition of absolute value:

  $|x|= \begin{cases} x& \text{if }x>0 \\ -x& \text{if }x<0 \\ 0& \text{if }x=0 \end{cases}$

  Solution

  Poor Jasmine. She didn't know to bring warm clothes.

  She assumed the temperature at her vacation spot would be $90^{\circ}$F. But what happened?

  A $30^{\circ}$F temperature difference was advertised.

  One possible solution: a temperature of $90^{\circ}$F because $90-30=60$.

  Unfortunately, there was no guarantee of a higher temperature. The temperature could also be colder than the home temperature.

  Let's solve the absolute value equation we set up earlier.

  $|x-60|=30$

  The definition of an absolute value equation is as follows:

  $|x|= \begin{cases} x& \text{if }x>0 \\ -x& \text{if }x<0 \\ 0& \text{if }x=0 \end{cases}$

  Splitting the equation above into two equations gives us:

  • $x-60=30$
  • $x-60=-30$

  Add $60$ to both sides of both equations to get two solutions:

  • $x=90^{\circ}$F
  • $x=30^{\circ}$F

  To figure out where she went wrong, Jasmine can spend her time inside the hotel solving absolute value equations.

• Solve the absolute value equation.

  Hints

  The definition of an absolute value is:

  $|x|= \begin{cases} x& \text{if }x>0 \\ -x& \text{if }x<0 \\ 0& \text{if }x=0 \end{cases}$

  Use opposite operations to solve equations.

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Solution

  Let's solve the absolute value equation: $|4x+20|=100$.

  Remember the definition for absolute value:

  $|x|= \begin{cases} x& \text{if }x>0 \\ -x& \text{if }x<0 \\ 0& \text{if }x=0 \end{cases}$

  According to the definition above, we write two equations:

  • $4x+20=100$
  • $4x+20=-100$

  Solve each equation:

  $\begin{array}{rcll} 4x + 20 &=& -100& \\ \color{#669900}{-20} && \color{#669900}{-20} &~~~~~|~ \text{Opposite Operations} \\ 4x &=& -120 \\ \color{#669900}{\div{4}} && \color{#669900}{\div{4}} &~~~~~|~ \text{Opposite Operations} \\ x &{=}& -30 & \end{array}$

  $~$

  $\begin{array}{rcll} 4x + 20 &=& 100& \\ \color{#669900}{-20} && \color{#669900}{-20} &~~~~~|~ \text{Opposite Operations} \\ 4x &=& 80 \\ \color{#669900}{\div{4}} && \color{#669900}{\div{4}} &~~~~~|~ \text{Opposite Operations} \\ x &{=}& 20 & \end{array}$

  We can check both solutions by plugging them into the corresponding equation:

  • $|4(-30)+20|=|-120+20|=|-100|=100$ $\surd$
  • $|4(20)+20|=|80+20|=|100|=100$ $\surd$

• Deduce the resulting rates.

  Hints

  For example:

  • The absolute value equation $|x-3|=4$ produces two equations.
  • $x-3=4$ and $x-3=-4$

  You can solve the equation above by adding $3$ to both sides of both equations to calculate: $x=7$ and $x=-1$.

  One car is faster than the other. Check your formula!

  Solution

  We know the top speed of Jasmine's old car, 75 mph. We also know that Jasmine's parents have promised a difference of 45 mph compared to her old car. We'll let $x$ be the unknown top speed of the new car. We can describe this situation by using an absolute value equation:

  $|x-75|=45$

  Because $|45|=|-45|$, we have to set up two equations - without the absolute value bars:

  $\begin{array}{rcll} x-75 & = & 45 \\ \color{#669900}{+75} & & \color{#669900}{+75} &~~~~~|~ \text{Opposite Operations} \\ x& = & 120\\ \end{array}$

  This is the faster car. Jasmine can race at a speed of 120 mph.

  $\begin{array}{rcll} x-75 & = & -45 \\ \color{#669900}{+75} & & \color{#669900}{+75} &~~~~~|~ \text{Opposite Operations} \\ x& = & 30\\ \end{array}$

  This is the slower car. Jasmine can putter around town at a speed of 30 mph.

• Analyze the absolute value equations.

  Hints

  An absolute value equation can generate zero equations or up to two equations depending on the value of the number on the right side of the equal sign.

  Look at the example $|x-7|=a$. The number of solutions depends on the value of $a$:

  $|x-7|=a\text{ has } \begin{cases} \text{two solutions}& \text{if }a>0 \\ \text{one solution}& \text{if }a=0 \\ \text{no solutions}& \text{if }a<0 \end{cases}$

  First rewrite each absolute equation so that you have the absolute value term on the left side of the equal sign and a constant on the right.

  Solution

  The number of solutions to an absolute value equation depends on the term on the right side:

  • There are two solutions if the right side is positive.
  • There is one solution if the right side is equal to zero.
  • There are zero solutions if the right side is negative.

  $\begin{array}{rclcrcl} &&&\mathbf{|2x+3|=2}&&& \\ 2x + 3 &=& 2& &2x+3&=&-2\\ \color{#669900}{-3} && \color{#669900}{-3} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-3} && \color{#669900}{-3}\\ 2x &=& -1 & &2x &=& -5\\ \color{#669900}{\div{2}} && \color{#669900}{\div{2}} &~~~|~ \text{Opposite Operations}~|~~~&\color{#669900}{\div{2}} && \color{#669900}{\div{2}} \\ x &=& -0.5 & & x &=& -2.5 \end{array}$

  Since the number on the right side of the equal sign is positive, there are two solutions.

  $~$

  $\begin{array}{rclcrcl} &&&\mathbf{|2x+3|+2=2}&&& \\ &&&\mathbf{|2x+3|=0}&&& \\ 2x + 3 &=& 0& &&& \\ \color{#669900}{-3} && \color{#669900}{-3} &~~~|~ \text{Opposite Operations}~|~~~ &&& \\ 2x &=& -3& &&& \\ \color{#669900}{\div{2}} && \color{#669900}{\div{2}} &~~~|~ \text{Opposite Operations}~|~~~ &&& \\ x &=& -1.5 & &&& \end{array}$

  Since the absolute value is equal to zero, there is only one solution.

  $~$

  $\mathbf{|2x+3|+5=2}$

  This equation is equivalent to $|2x+3|=-3$. Notice the number on the right side of the equal sign is negative. Since this is not possible, the equation has no solution.

  $~$

  $\begin{array}{rclcrcl} &&&\mathbf{-|2x+3| + 5=2}&&& \\ &&&\mathbf{-|2x+3| = -3}&&& \\ &&&\mathbf{|2x+3| = 3}&&& \\ 2x + 3 &=& 3& &2x+3&=&-3\\ \color{#669900}{-3} && \color{#669900}{-3} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-3} && \color{#669900}{-3}\\ 2x &=& 0 & &2x &=& -6\\ \color{#669900}{\div{2}} && \color{#669900}{\div{2}} &~~~|~ \text{Opposite Operations}~|~~~&\color{#669900}{\div{2}} && \color{#669900}{\div{2}} \\ x &=& 0 & & x &=& -3 \end{array}$

  Since the number on the right side of the equal sign is positive, there are two solutions.

  $~$

  $\mathbf{-|2x+3|=2}$

  This equation is equivalent to $|2x+3|=-2$. Notice the number on the right side of the equal sign is negative. Since this is not possible, the equation has no solution.

• Explain why absolute value equations can have two solutions.

  Hints

  Remember the definition for absolute value:

  $|x|= \begin{cases} x& \text{if }x>0 \\ -x& \text{if }x<0 \\ 0& \text{if }x=0 \end{cases}$

  An absolute value equation is always equal to zero or a positive number.

  Solution

  This is an example of an absolute value equation. Notice the variable is located inside the absolute value bars.

  • $|x-60|=30$

  Because $|30|=|-30|$, we have to solve the equations $x-60=30$ and $x-60=-30$. Adding $60$ to both sides of both equations, we calculate the solutions $x=90$ and $x=30$.

  Therefore, absolute value equations can have as many as two solutions.

  Can an absolute value equation have zero solutions? Sure! If we look at $|x-2|=-6$, we can conclude that this equation has no solution. Why? The absolute value of any number is always equal to a positive number or zero.

  Therefore, it is possible to have an absolute value equation with zero solutions.

  Can an absolute value equation have only one solution? This is also possible. Just look at the following example $|x-2|=0$. According to the definition of an absolute value equation, we get one equation: $x-2=0$, since zero isn't positive nor is it negative $x=2$ is the only possible solution.

• Solve the absolute value equations.

  Hints

  Manipulate the equations to show the absolute value term only on the left side of the equal sign and the constant on the other.

  Use opposite operations to manipulate the equations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  You can also check the each solution by plugging it into the absolute value equation.

  Solution

  To solve absolute value equations:

  1. Rewrite each absolute value equation to show the absolute value term on the left side of the equal sign and the constant on the right side.
  2. Leave off the absolute value bars.
  3. Solve each equation by using opposite operations.

  Equation 1: $\mathbf{2|x+1|-3=5}$

  $\begin{array}{rclcrcl} &&&\mathbf{2|x+1|-3=5}&&& \\ &&&\mathbf{2|x+1|=8}&&& \\ &&&\mathbf{|x+1|=4}&&& \\ x + 1 &=& 4& &x+1&=&-4\\ \color{#669900}{-1} && \color{#669900}{-1} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-1} && \color{#669900}{-1}\\ x &=& 3 & &x &=& -5 \end{array}$

  $~$

  Equation 2: $\mathbf{-\frac12|x+4|=-2}$

  $\begin{array}{rclcrcl} &&&\mathbf{-\frac12|x+4|=-2}&&& \\ &&&\mathbf{\frac12|x+4|=2}&&& \\ &&&\mathbf{|x+4|=4}&&& \\ x + 4 &=& 4& &x+4&=&-4\\ \color{#669900}{-4} && \color{#669900}{-4} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-4} && \color{#669900}{-4}\\ x &=& 0 & &x &=& -8 \end{array}$

  $~$

  Equation 3: $\mathbf{|2x+4|=6}$

  $\begin{array}{rclcrcl} &&&\mathbf{|2x+4|=6}&&& \\ 2x + 4 &=& 6& &2x+4&=&-6\\ \color{#669900}{-4} && \color{#669900}{-4} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-4} && \color{#669900}{-4}\\ 2x &=& 2 & &2x &=& -10 \\ \color{#669900}{\div2} && \color{#669900}{\div2} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{\div2} && \color{#669900}{\div2}\\ x &=& 1 & &x &=& -5 \\ \end{array}$

  $~$

  Equation 4: $\mathbf{4-|2x+3|=-7}$

  $\begin{array}{rclcrcl} &&&\mathbf{4-|2x+3|=-7}&&& \\ &&&\mathbf{-|2x+3|=-11}&&& \\ &&&\mathbf{|2x+3|=11}&&& \\ 2x + 3 &=& 11& &2x+3&=&-11\\ \color{#669900}{-3} && \color{#669900}{-3} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{-3} && \color{#669900}{-3}\\ 2x &=& 8& &2x&=&-14\\ \color{#669900}{\div2} && \color{#669900}{\div2} &~~~|~ \text{Opposite Operations}~|~~~ &\color{#669900}{\div2} && \color{#669900}{\div2}\\ x &=& 4 & &x &=& -7 \\ \end{array}$