Introduction to Absolute Value

Introduction to Absolute Value exercise

• Describe absolute value.

  Hints

  Note

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

  Negative numbers are smaller than positive numbers.

  Absolute values are always positive numbers.

  Solution

  As far away from sea level as possible:

  • could be an elevation such as the peak of Mount Everest, or
  • could as well the deepest point of the Mariana Trench.

  So from sea level

  • it's about 29000 feet to the peak of Mount Everest and
  • about -36000 feet to the deepest point of the Mariana Trench.

  Obviously $-36000<29000$.

  But, to determine the distance from sea level, we have to look at the absolute values:

  • $|29000|=29000$ and
  • $|-36000|=36000$.

  Note that an absolute value is always a positive number.

  We get the largest distance from sea level:

  $36000>29000$.

  Therefore, Lara and Philip will spend their next holiday in the Mariana Trench.

• Determine the absolute value of the numbers.

  Hints

  Note

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

  The absolute value is always positive. So, the negative of an absolute value is always negative.

  Working with more complicated expressions: First, solve the expression that is between the absolute value bars, and then find the absolute value of the simplified expression.

  Solution

  The absolute value of a positive number is simply equal to the number: $|3|=3$.

  The absolute value of a negative number is equal to the opposite of that number and is always positive: $|-5|=5$.

  The absolute value of zero is zero: $|0|=0$.

  What is the negative of an absolute value?

  • $-|3|=-(3)=-3$ as well as
  • $-|-5|=-(5)=-5$

  So, it's a negative number.

  And to solve more complicated expressions, simplify the expression within the bars first then find the absolute value of the simplified expression:

  • $|3+(-5)|$: We have to calculate $|3+(-5)|=|3-5|=|-2|=2$.
  • $|2\times 3+(-5)|=|6+(-5)|=|6-5|=|1|=1$.

• Determine the winner of the absolute value game.

  Hints

  The absolute value of a positive number is the number itself: $|32|=32$.

  The absolute value of a negative number is the opposite of the number: $|-64|=64$.

  The absolute value of zero is zero: $|0|=|0|$.

  If you have to determine the absolute value of a more complicated expression, first simplify the expression.

  Use the Distributive Property:

  $a\times(b+c)=a\times b+ a\times c$.

  Solution

  The absolute value of a positive number is the number itself: $|32|=32$. The absolute value of a negative number is the opposite of the number $|-64|=64$. The absolute value of zero is zero: $|0|=|0|$.

  If we have to determine the absolute value of a more complicated expression, first simplify the expression.

  For each of the following examples, we will determine the the absolute value. Afterwards we 'll be able to decide who wins the game.

  1. $|-23|=23$ and $|46|=46$. So $|-23|<|46|$. Philip wins.
  2. $|-46|=46$ and $|23|=23$. So $|-46|>|23|$. Lara wins.
  3. $|-23|=23$ as well as $|23|=23$. So $|-23|=|23|$. Draw.
  4. $|46|46$ as well as $|-46|=46$. So $|46|=|-46|$. Draw.
  5. $|23+46|=|69|=69$ and $|23-46|=|-23|=23$. So $|23+46|>|23-46|$. Lara wins.
  6. $|-23-46|=|-69|=69$ and $|46-23|=|23|=23$. So $|-23-46|>|46-23|$. Lara wins.
  7. $|-23+46\times2|=|-23+92|=|69|=69$ and $|3\times (-23)|=|-69|=69$. So $|-23+46\times2|=|3\times (-23)|$. Draw.
  8. $|2\times(-23+46)|=|2\times(-23)|=|-46|=46$ and $|2|\times(|-23|+|46|)=2\times(23+46)=2\times 69=138$. So $|2\times(-23+46)|<|2|\times(|-23|+|46|)$. Philip wins.

  Lara wins three times while Philip only wins twice. So he has to pay for lunch today.

• Decide which terms have the same absolute value.

  Hints

  The absolute value of a positive number is simply equal to the number.

  The absolute value of a negative number is the opposite of the number.

  First, determine the value of the expression between the absolute value bars.

  For example:

  $|2\times(-2+3)|=|2\times 1|=|2|=2$.

  Solution

  Here is cheat sheet for absolute value:

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

  Here we have different expressions using absolute value:

  1. $|-6+9|=|3|=3$
  2. $|6|-|-9|=6-9=-3$
  3. $-|9-6|=-|3|=-3$
  4. $|-9+6|=|-3|=3$
  5. $|-9|-|-6|=9-6=3$
  6. $-|-9|+|-6|=-9+6=-3$

• Decide which is further from sea level.

  Hints

  For example $|-3|=3$ as well as $|3|=3$.

  The absolute value on a number line is the distance of any number to the zero point.

  Look at the following example:

  • $-6<4$, but
  • $|-6|=6>4$.

  Solution

  Let's have a look at the information we're given:

  • From sea level, it's about 29000 feet to the peak of Mount Everest and
  • about -36000 feet to the deepest point of the Mariana Trench.

  To decide which is further away from sea level, we have to take a look at the absolute values:

  • $|29000|=29000$ and
  • $|-36000|=36000$.

  Keep in mind that absolute values are always positive numbers.

  $36000>29000$.

  So that's the decision: They will go to the Mariana Trench.

• Find the absolute values.

  Hints

  The number of steps is always positive while the position on the number line can be negative.

  For example: On this number line you can see two steps to the left, the red arrow, and one to the right, the blue arrow.

  The position is $-1$ and the number of steps is $|-2|+|1|=2+1=3$.

  Four steps to the left gives us

  • the position $-4$ and
  • $|-4|=4$ steps.

  Solution

  The number line on the right shows

  • Susan taking three steps to the left and then another step to the left.
  • The resulting position is $-4$ and
  • the number of steps is $|-3|+|-1|=3+1=4$.

  In a same manner, we can model the other examples:

  • Seven steps to the right give us the position $7$ and five steps to the left the position $2$. The number of steps can be determined by $|7|+|-5|=7+5=12$.

  • Eight steps to the left give us the position $-8$ and twelve steps to the right the position $4$. The number of steps can be determined by $|-8|+|12|=8+12=20$.