Absolute Value Equations and Inequalities Notes Absolute Value Equations When solving equations you can have at most 1 solution because the variable is a linear term. When we are solving absolute value equations that changes just a little bit. The absolute value of a number is the distance that the number is away from 0. For example, the absolute value of 4 is 4 because it is 4 units away from 0; the absolute value of -­‐4 is also 4 because it is also 4 units away from 0. Mathematicians represent the absolute value operation by . So 4 = 4 !"# −4 = 4. So lets consider the following equation…. Solve for x ! = 7 When looking at this equation, you will know that the value of x could be 7 because 7 = 7 and you also know the value of x could be -­‐7 because −7 = 7 You can always check your answers. Check your answers 7 = 7 −7 = 7 7 = 7 7 = 7 True Statements Therefore, the solutions to that absolute value equation are 7 and -­‐7. There are 2 answers. So that one was fairly easy J Lets try something a little bit more difficult −8 − 2 5! − 7 = −14 This one requires a little bit more work because the absolute value signs are not by themselves and there are numbers inside the absolute value signs as well. We will solve this equation after we look at the following steps and do a few examples. Now, lets think about this, will there always be 2 solutions? Could there be 1 or no solution? We will find out throughout this set of notes. 1 Absolute Value Equations Given that a, b, and c are numbers and x is a variable, |!" + !| = ! is equivalent to !" + ! = −! !" + ! = ! Steps to solve absolute value equations 1. Isolate the absolute value expression. 2. Rewrite the equation and get rid of the absolute value signs. (Set the expression inside the absolute value signs equal to the negative of the “c” value and another equation equal to the positive of the “c” value. 3. Solve for the variable in both equations. 4. Both x values are solutions to the absolute value equation. Example 1 You don’t have to isolate the absolute value expression because it is already isolated in the original problem. Solve for x. ! − 5 = 9 ! − 5 = 9 ! − 5 = −9 ! = 14 ! = −4 Check your answers 14 − 5 = 9 −4 − 5 = 9 9 = 9 −9 = 9 9 = 9 9 = 9 True Statements The solutions are x = 14 and x = -­‐4. 2 Example 2 Solve for x −6! + 5 = 8 −6! + 5 = −8 − 6! + 5 = 8 −6! = −13 − 6! = 3 13
−1
! =
! =
6
2
Check your answers 13
−1
−6
+ 5 = 8 −6
+ 5 = 8 6
2
−13 + 5 = 8 3 + 5 = 8 −8 = 8 8 = 8 8 = 8 8 = 8 True Statements You don’t have to isolate the absolute value expression because it is already isolated in the original problem. !"
!!
The solutions are ! = ! !"# ! = ! Example 3 Solve for x You do have to isolate the absolute value −8 − 2 5! − 7 = −14 expression because there numbers on the outside of the absolute value expression. You will have to add 8 and divide by -­‐2 BEFORE you rewrite the expression. −2 5! − 7 = −6 5! − 7 = 3 5! − 7 = −3 5! − 7 = 3 5! = 4 5! = 10 !
! = ! ! = 2 Check your answers 4
−8 − 2 5
− 7 = −14 − 8 − 2 5(2) − 7 = −14 5
−8 − 2 4 − 7 = −14 − 8 − 2 10 − 7 = −14 3 −8 − 2 −3 = −14 − 8 − 2 3 = −14 −8 − 2(3) = −14 − 8 − 2(3) = −14 −8 − 6 = −14 − 8 − 6 = −14 −14 = −14 − 14 = −14 !
The solutions are ! = ! !"# ! = 2. Example 4 You do have to isolate the absolute value Solve for x expression because there numbers on the 6 + 5 2! − 5 = −8 outside of the absolute value expression. You will have to subtract 6 and divide by 5 BEFORE you rewrite the expression. 5 2! − 5 = −14 −14
2! − 5 =
5
Now lets examine our statement. If claims that the absolute value of an expression is a negative number, this is not possible because the absolute value measures a distance, and distance (absolute value) can’t be negative. So there isn’t a solution to this equation. There is no solution to this equation. Example 5 Solve for x You do have to isolate the absolute value 5 − 4 −2! − 8 = 5 expression because there numbers on the outside of the absolute value expression. You will have to subtract 5 and divide by -­‐4 BEFORE you rewrite the expression. −4 −2! − 8 = 0 −2! − 8 = 0 As we look at the above statements, and you start to rewrite the expressions there is not a -­‐0 so this can change to just one statement. −2! − 8 = 0 −2! = 8 ! = −4 4 Check you solution 5 − 4 −2(−4) − 8 = 5 5 − 4 8 − 8 = 5 5 − 4 0 = 5 5 − 4 0 = 5 5 − 0 = 5 5 = 5 True Statements There is 1 solution to this absolute value equation and it is x= -­‐ 4. As you can see in the above examples, absolute value equations can have 0, 1, or 2 solutions. Now lets look at what happens when we start solving absolute value inequalities. Absolute Value Inequalities Now that we have talked absolute value equations, lets look at absolute value inequalities. Lets look at this example. Example 6 ! < 5 This states that the absolute value of a value “x” has to be smaller than 5. We know that -­‐5 and 5 have an absolute value of 5 so the values that are between these two values should have absolute values smaller than 5. Look at the number line below. The numbers outlined are solutions to ! < 5 Any x value that is between -­‐5 and 5 would have an absolute value smaller than 5. We can write this solution to be −5 < ! < 5 . (Remember from the lesson on linear inequalities, that this is an “AND” statement, meaning that x has to be greater than –5 AND smaller than 5. ) What if the inequality changed to the following Example 7 5 ! ≤ 5 This means that the absolute value has to be smaller than 5 or it can be equal to 5. So the solutions slightly change. Look at the number line below. The numbers that are outlined are solutions to ! ≤ 5 In this example the -­‐5 and the 5 are included in the solutions because the absolute value can equal 5. We can write this solution to be −5 ≤ ! ≤ 5 . (Remember from the lesson on linear inequalities, that this is an “AND” statement, meaning that x has to be greater than or equal to –5 AND smaller than or equal to 5. ) Now lets change the inequality sign and see what happens. Example 8 ! > 5 This states that the absolute value of a value “x” has to be greater than 5. We know that -­‐5 and 5 have an absolute value of 5. The values that larger than 5 have an absolute value greater than 5 and the values that are smaller than -­‐5 will have an absolute value that is greater than 5. Look at the number line below. The numbers that are outlined are solutions to ! > 5 This shows that any value smaller than -­‐5 OR bigger than 5 would have an absolute value of greater than 5. We can write the solution as ! < −5 !" ! > 5. Example 9 ! ≥ 5 This states that the absolute value of a value “x” has to be greater than or equal to 5. We know that -­‐5 and 5 have an absolute value of 5. The values that larger than 5 have an absolute value greater than 5 and the values that are smaller than -­‐5 will have an absolute value that is greater than 5. Look at the number line below. The numbers that are outlined are solutions to ! ≥ 5 6 This shows that any value smaller than or equal to -­‐5 OR bigger than of equal to 5 would have an absolute value of greater than or equal to 5. We can write the solutions as ! ≤ −5 !" ! ≥ 5. Absolute Value Inequalities Given that a, b, and c are numbers and x is a variable, |!" + !| < ! is equivalent to −! < !! + ! < ! |!" + !| ≤ ! is equivalent to −! ≤ !" + ! ≤ ! |!" + !| > ! is equivalent to !" + ! < −! !" !" + ! > ! |!" + !| ≥ ! is equivalent to !" + ! ≤ −! !" !" + ! ≥ ! Steps to solve an Absolute Value Inequality 1. Isolate the absolute value expression. 2. Move the absolute value expression to the left side of the inequality. 3. Rewrite the absolute value inequality to an “AND” or an “OR” statement (Look at the table above) 4. Solve for x with the same rules used to solve for x with compound inequalities. Please note that the absolute value expression is on the left side of the inequality…..if the inequality is on the right side, rewrite the inequality so that its on the left and then rewrite the inequality based on the table above. If the absolute value expression is on the left, the signs < & ≤ make an “AND” compound inequality. If the absolute value expression on the left , the signs > & ≥ make an “OR” compound inequality. 7 Example 10 Solve for x and graph the solutions. Since this inequality is a less than inequality, we can rewrite this to ! − 6 < 13 be an “AND” compound inequality. −13 < ! − 6 < 13 −7 < ! < 19 This means that any value between -­‐7 and 19 that you substitute in for x in the absolute value inequality, ! − 6 < 13, will give a true statement. So the solutions are −7 < ! < 19 Example 11 Solve for x and graph the solutions. Since this inequality is a greater than or equal to −6! − 8 ≥ 12 inequality, we can rewrite this to be an “OR” compound inequality. −6! − 8 ≤ −12 !" − 6! − 8 ≥ 12 −6! ≤ −4 !" − 6! ≥ 20 4
−20
! ≥ !" ! ≤
6
6
2
−10
! ≥ !" ! ≤
3
3
!
This means that any x-­‐value that is greater than or equal to ! OR an x-­‐value that is !!"
smaller than or equal to !
when you substitute it into −6! − 8 ≥ 12 will give !
you a true statement. So the solutions are ! ≥ ! !" ! ≤
!!"
!
8 Example 12 Solve for x and graph the solutions. Since this absolute value inequality 5 − 8 ! − 1 < 3 is not isolated you have to subtract −8 ! − 1 < −2 the 5 and divide by a -­‐8. 1
REMEMBER, when you divide by a ! − 1 > 4
negative number you change the direction of the inequality sign. Now is a greater than sign so it becomes an “OR” compound inequality. −1
1
!−1<
!" ! − 1 > 4
4
3
5
! < !" ! > 4
4
!
!
This means that any x value that is less than ! OR greater than !, when you substitute those values in for x in 5 − 8 ! − 1 < 3 you will get a true statement. So !
!
the solutions are ! < ! !" ! > ! Example 13 Solve for x and graph the solutions. 9 + 10 8! ≤ 59 10 8! ≤ 50 8! ≤ 5 −5 ≤ 8! ≤ 5 −5
5
≤ ! ≤ 8
8
Since this absolute value inequality is not isolated you have to subtract the 9 and divide by 10. Now is a less than or equal to sign it becomes an “AND” compound inequality. 9 !!
This means that an x value that is greater than or equal to ! and smaller than or !
equal to !, when you substitute those values in for x in 9 + 10 8! ≤ 59 you will get !!
!
a true statement. So the solutions are ! ≤ ! ≤ !. Absolute Value Inequality Application Word Problems Now that you have seen the process to solve absolute value inequalities where can we use these in real world situations? When solving word problems using absolute value inequalities here are the steps you need: Step 1: Define a variable. Step 2: Write an absolute value inequality. Step 3: Solve the absolute value inequality. Step 4: State your answer. Example 14 Sarah has $15. She knows that Blake has some money and it varies by at most $5 from the amount of her money. Write an absolute value inequality that represents this scenario. What are the possible amounts of money that Blake can have? When you are think about this scenario you know that Blake could have more or less money than Sarah does…but it can only vary by at most $5, meaning the amount of money that he has could be $10 all the way up through $20. So logically we already know the answer. But what absolute value inequality would represent this scenario? So lets follow the steps. 10 Step 1: define a variable ! = !ℎ! !"#$%& !" !"#$% !"#$% ℎ!" Step 2: If you take Blake’s money total, and you subtract Sarah’s total of 15…there shouldn’t be a difference or 5 dollars….so lets think When thinking ! − 15 there are 2 ! − 15 scenarios Scenario 1: Blake has more money than Sarah If ! = 18 then 18 − 15 = 3 meaning that he doesn’t vary more than 5 dollars. ! − 15 ≤ 5 this takes care of the negative values because the absolute value of the Scenario 2: Blake has less money than negative values has to be less than or equal to 5 Sarah to match the scenario. If ! = 13 then 13 − 15 = −2 the negative means he has less money than Sarah but Step 3: −5 ≤ ! − 15 ≤ 5 he still doesn’t vary more than $5 10 ≤ ! ≤ 20 So how do we take into consideration both Step 4: The absolute value inequality that scenarios?....YOU guessed it…Absolute matches this scenario is ! − 15 ≤ 5 . The value amount of money that Blake could have is 10 ≤ ! ≤ 20 , which means that he could have any amount from $10 up to $20. Looking at the absolute value inequalities in general terms to solve word problems. You will have the variable that you defined minus the “average” or middle of the range, following by an inequality sign (determined by the question), and then the amount that it varies. Example 15 a) A company is making mirrors. Each mirror needs to be 18 inches high. Each mirror can vary by at most 0.01 inch. Write an absolute value inequality that represents the situation and allows you to answer the question; what are the acceptable heights of this company’s mirrors? Step 1: ℎ = ℎ!"#ℎ! !" !"##$# Step 2: ℎ − 18 ≤ 0.01 Step 3: −0.01 ≤ ℎ − 18 ≤ 0.01 17.99 ≤ ℎ ≤ 18.01 11 Step 4: The absolute value inequality that represents this scenario is ℎ − 18 ≤
0.01. The acceptable heights of the mirrors are 17.99 ≤ ℎ ≤ 18.01. This means, any mirror 17.99 inches up to 18.01 inches is acceptable. Now lets change it up to find the unacceptable heights of mirror. Example 16 b) A company is making mirrors. Each mirror needs to be 18 inches high. Each mirror can vary by at most 0.01 of an inch. Write an absolute value inequality that represents the situation and allows you to answer the question; what are the unacceptable heights of this company’s mirrors? Step 1: ℎ = ℎ!"#ℎ! !" !"##$# Notice that the inequality signed Step 2: ℎ − 18 > 0.01 changed from example 15 to 16 because we changed from acceptable heights to unacceptable heights. And the unacceptable heights would have to vary MORE THAN 0.01 inches. Step 3: ℎ − 18 < −0.01 !" ℎ − 18 > 0.01 ℎ < 17.99 !" ℎ > 18.01 Step 4: The absolute value inequality that represents this scenario is ℎ − 18 >
0.01. The unacceptable heights of the mirrors are ℎ < 17.99 !" ℎ > 18.01 . This means, any mirror below 17.99 inches and anything above 18.01 inches isn’t acceptable. Example 17 The weather forecast for today states that the average temperature is going to be 55°! and the temperature is going to vary by at most 8°!. Write an absolute value inequality that represents this scenario. What are the possible temperatures for today? Step 1: ! = !"##$%& !"#$"%&!'%" !" °! Step 2: ! − 55 ≤ 8 Step 3: −8 ≤ ! − 55 ≤ 8 47 ≤ ! ≤ 63 Step 4: The absolute value inequality that represents this scenario is ! − 55 ≤ 8. The possible temperatures of today are 47 ≤ ! ≤ 63. This means, temperatures can range from 47℉ !" 63℉ Example 18 12 A company sells bags of oranges. Each bag of oranges should weight 8.5 pounds. Each bag can vary by at most 1.5 pounds. The company ships their oranges with 10 bags of oranges to a box. What are the unacceptable weights of 1 box of oranges? Step 1: ! = !"#$ℎ! !" 1 !"# !" !"#$%&' Step 2: ! − 85 > 15 Step 3: ! − 85 < −15 !" ! − 85 > 15 ! < 70 !" ! > 100 Step4: The unacceptable weight for each box is ! < !" !" ! > !"" , which means the boxes shouldn’t weight less than 70 pounds or more than 100 pounds. 13