Absolute Value Lesson, Examples, & Real World Problems

Solving/Graphing Absolute Value Inequalities & Equations

Since absolute value = amount can go either way on number line, split absolute value inequality into 2 inequalities, dropping absolute value & solving. On the second equation, flip inequality and sign of the number to address the movement the opposite way.

Multiple Choice

What's a way to define absolute value?

• The value of a number

• The opposite of a number

• The distance of a number from zero

• The multiplicative inverse of a number

Multiple Choice

Solve:

|x + 3| > 8

• x > 5 or x < -11

• x > 5

• all real numbers

• no solution

This is a > inequality that will result in 2 sets of solutions, called a union.

When working an absolute value, you create two separate inequalities, flipping the signs of both the inequality and the integer.

Since the original inequality is greater than, we know this is an OR compound inequality that will have a union of two sets of answers rather than an intersection of answers. In a union, the answers could be those over there OR those over the other way. Do you remember how it is written in interval notation?

 $\left(-\infty,-11\right)\cup\left(5,\ \infty\right)$    ALSO - notice that the answer above is not written in order. When you write your inequality, the numbers should be in order from least to greatest, moving left to right. *INTERVAL NOTATION - Notice the parentheses. You ALWAYS use them with infinity and with < or >. 

Multiple Choice

Write the inequality in interval notation.

• [-4, -1)

• (-4, -1)

• (-4, -1]

• [-4, -1]

An Absolute Value EQUATION also breaks into two equations. We just change the sign of the second one, leaving the = alone.

PAY CLOSE ATTENTION TO SIGNS!!!!!!

Absolute value EQUATION Example

The answers can also be in set notation, which looks like {3, 6} It is a SET of answers, not an interval.

The answer set on a number line for an EQUATION will just be the dots. No arrows, because there are no other possible answers like in an inequality.

x=3, x=6 or {3, 6} (set notation)

It's answer set isn't an interval so you don't use interval notation. You will see either of these:

a. x=3, x=6 b. x =3,6 c. {3, 6}, which is set notation. The two answers that are possible for the x value is a set of numbers. 3 and 6. Not an interval. Make sense?

Multiple Select

What value(s) of x will satisfy the equation? Choose all that apply.

 $\left|2x\right|=6-x$  

• -6

• -4

• 0

• 2

Multiple Choice

Solve |x – 12| - 3 = 11.

• x = -2 and x = -26

• x = -2 and x = 26

• x = 3 and x = 25

• x = 4 and x = -7

Multiple Choice

Solve |x – 12| - 3 = 11. The answers are x = -2 and x = 26. How is that written in set notation? (IT ISN'T AN INTERVAL, just a set of two #s)

• {x=2}

• {-2, 26}

• [-2, 26)

• [-2, 26]

Example ABS VALUE EQUATION:

Isolate the absolute value like you would the X. Then, it gets a little different. They split up because an absolute value gives you two answers from two directions.

Multiple Choice

Solve for x
l x−6 l + 4= 10

• x = 12 and x = 0
• x = 3 and x = -3
• x = -12 and x = 0
• No Solution

Multiple Choice

Solve.

 $2\left|x-25\right|+4\ =\ 12$  

•  $\left\{29\right\}$  

•  $\phi$  

•  $\left\{21,\ 29\right\}$  

•  $\left\{17,\ 29\right\}$  

Multiple Choice

−2|−2r − 4| = -12

• x = 5 and x = 1

• x = -5 and x = -1

• x = -5 and x = 1

• No Solution

Real world absolute value applications: Sea Level

• It's important you understand how these "real world" absolute value operations work. This is not only real life, but understanding these concepts bleeds over to MANY other math skills, especially in trigonometry.

• SEA LEVEL is considered 0 (zero) on your number line. The starting point.

Multiple Choice

The top of a mountain is 2,450 ft above sea level. A shipwreck was discovered 310 feet below sea level. What is the distance between the mountain top and the shipwreck?

• 2150 feet

• 310 feet

• 2760 feet

• 2450 feet

Multiple Select

Natural Bridge in VA is 215 feet above ground level while neary Luray Caverns is 260 feet below ground. Which two absolute value expressions could be used to find the distance between the top of Natural Bridge and the bottom of Luray Caverns?

• |215 - -260|

• |215 - 260|

• |-260 - 215|

Multiple Choice

The platform of an oil rig is at sea level. The tallest structure on the platform is 120 feet tall. The oil rig structure under the platform goes down 1,250 feet below sea level to attach to the ocean floor. What is the distance from the top of the tallest structure to where the structure attaches to the ocean floor?

• 1,370 feet

• 1,130 feet

• 1,125 feet

Homework problem from last week! The amount a value can move either way is what goes on the "outside" of the equality $\left|x-55\right|\le3$  

Absolute Value & Distance

• Absolute Value is a way to eliminate negative numbers. For example, if you walk to school 2 miles, it's a 2 mile walk.

• However, when you walk back home it's not a negative 2 (-2) walk! That's an example of absolute value.

Multiple Choice

Sam's house is 14 blocks from the school. Emily lives 6 blocks from Sam. Which equation represents the location of Emily's house in relation to the school?

• |x + 6| = 10
• |x – 6| = 14
• |x – 14| = 6
• |x + 14| = 6

Multiple Select

A house is built at the top of a 49 foot hill. Another house is built in a valley 60 feet below the base (bottom) of the hill. What is the zero point in this problem?

• The top of the hill.

• The base of hill.

• The house in the valley.

Multiple Select

The diving board is 25 ft. above the water level of the pool. If the Daring Diver dives off of the board and goes down to 12 feet below water level in the pool, which two absolute value expressions could be used to find how many feet the Daring Diver travelled?

• |12 - 25|

• |25 - (-12)|

• |-12 - 25|

• |-12 - (-25)|

Multiple Choice

The diving board is 25 ft. above the water level of the pool. If the Daring Diver dives off of the board and goes down to 12 feet below water level in the pool. How many feet did the Daring Diver travel?

• 37 feet

• 13 feet

• -12 feet

• 25 feet

Multiple Choice

A diver was 40 feet below the surface of the water. A fisherman was casting off a dock right above the diver that was 12 feet above the surface of the water. How far apart are the diver and the fisherman?

• 28 feet
• -28 feet
• 52 feet
• -52 feet

Multiple Choice

The lowest temperature on Thursday was -20^o C.  The lowest temperature on Saturday was -12^o C.  What was the difference between the lowest temperatures?  

• 13^{o C}
• 32^o C
• 8^o C
• 9^o C

Multiple Choice

The first play of a football game resulted in a loss of 12 yards.  Then a penalty resulted in another loss of 5 yards.  What is the total loss or gain?

• a gain of 7 yards
• a loss of 7 yards
• a gain of 17 yards 
• a loss of 17 yards

Multiple Select

The high temperature in Fort Lauderdale was 98⁰ F, while the highest temperature in Greenland was -12⁰ F. Which two absolute value expressions could be used to find the difference between the two temperatures?

• |-12 - 98|

• |98 - 12|

• |98 - - 12|

Multiple Choice

Which compound inequality is equivalent to the absolute value inequality:

|b|> 6

• -6 > b AND b > 6

• b > -6 OR b < 6

• -b < 6 AND b > 6

• b < -6 OR b > 6

Multiple Choice

Which answer below represents the solution absolute value inequality:

|b|> 6 in interval notation?

• 
  

  $\left(-\infty,-6\right)\cup\left(6,\infty\right)$

• 
  

  $\left(-\infty,6\right)\cap\left(6,\infty\right)$

• $\left(-\infty,-6\right)\cup\left[6,\infty\right]$

• b < -6 OR b > 6

Multiple Choice

Express the following interval in a compound inequality.

• -7 ≤ x ≤ 15

• -7 < x ≤ 15

• -7 ≤ x < 15

• -7 < x < 15

Multiple Choice

Write the interval notation for the graph?

• [-3, 2)

• -3<x<2

• (-∞, -3] U (2, ∞)

• x > 2 or x < -3

Multiple Choice

| x - 6 | < 4
Is this an "and"  or an "or" problem?

• and
• or

Multiple Choice

Which compound inequality involves >,≥ when solving absolute value inequalities?

• and

• or

Multiple Choice

| 2x + 12 | < 8
Which is the correct first step?

• 2x + 12 < 8 and 2x+12 >-8
• 2x + 12 < 8 or 2x+12 >-8
• 2x + 12 < 8 and 2x+12 <-8
• 2x + 12 < 8 or 2x+12 >-8

Multiple Choice

Solve and graph:

• A

• B

• C

• D

Multiple Choice

Solve:
3 | 2x + 7| - 7 < -4

• x < -3 and x > -5/3
• x < -3 and x > -4
• all real numbers
• no solution

Multiple Choice

After you split into two inequalities, what would they look like? 

• x + 7 > 1        
     
  x + 7 > -1
• x + 7 > 1
• x + 7 > 1
  X + 7 < -1

Multiple Choice

What is the correct way to write the answer? 

• x = -2 AND x = 6
• x = -2 OR x = 6

Multiple Choice

Can you write x = -2 OR x = 6 in interval notation?

• Yes. This is an inequality that has multiple answers and can be written as an interval, or union or intersection of answers.

• No. This is an equation that has only a certain set of possible solutions and can only be written in set notation.

Multiple Choice

Your have money in your wallet, but you don't know the exact amount. When a friend asks you, you say that you have 50 dollars give or take 15. Write an absolute value equation to model this situation.

• |x-50|=15

• |x+15|=50

• |50+x|=15

• No Solution

Multiple Choice

A company sells 20-ounce bottles of sports drink. The actual volume of liquid in the bottle must be within 0.03 ounces. What equation can be solved to find the minimum and maximum volume of liquid in a bottle.

• |v|-20=0.03
• |v-20|=0.03
• |v|-0.03=20
• |v-0.03|=20

Multiple Choice

The street built in the city must be 25 feet in width with a tolerance of 0.5 feet. Streets that are not within the tolerated widths must be repaired. Which of the following inequalities can be used to assess which streets are within tolerance? (W is the width of the street) .

• |W - 25| ≤ .5

• |W - .5| ≤ 25

• |W - 25| ≥ .5

• |W - .5| ≥ 25

Multiple Choice

The radius of the gears produced at a factory must be 6 inches in length with a tolerance of 0.1 inches. The gears with radius beyond the tolerated lengths will be thrown away. Which of the following inequalities can be used to assess which gears are eligible? (x is the length of the radius)

• |x – 6| ≤ 0.1
• |x – 6| ≥ 0.1
• |x – 0.1| ≤ 6
• |x – 0.1| ≥ 6

Multiple Choice

John is looking for a job after graduation. The salary which he is satisfied with must be $3000 with a tolerance of $500. Which of the following inequalities can be used to assess which if his salary is tolerable? (m is the measure of the salary)

• |m - 500| ≤ 3000
• |m - 500| ≥ 3000
• |m - 3000| ≥ 500
• |m - 3000| ≤ 500

Multiple Choice

At the TX Design Company, the p.21 average starting salary for a new designer is $33,600, but the actual salary could differ from the average by as much as $1,560. Which inequality could be used to determine if a salary, x, falls within this range?

• |x ­- 1560| ≤ 33600

• x ­- 33600 ≤ 1560

• |x ­- 33600| ≥ 1560

• |x -­ 33600| ≤ 1560

Multiple Choice

An essay contest requires that essay entries consist of 500 words with an absolute deviation of at most 30 words. Write an absolute value inequality to represent this.

• |w + 500| > 30

• |w - 500| ≤ 30

• |w - 30| ≥ 500

• w - 30 < 500

Multiple Choice

A candy manufacture puts 240 pieces of candy into each bag. The bag can be off by 8 pieces. Write an absolute value inequality that displays the acceptable amount of candy pieces that can be in a bag.

Write an inequality to model this situation.

• |x - 8| ≤ 240

• x + 240 ≥ 8

• |x - 240| ≤ 8

• |x - 240| > 8