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Solving Absolute Value Equations

Solving Absolute Value Equations. What is Absolute Value?. Absolute Value : The absolute value of a number is the number of units it is from 0 on the number line . (Distance from 0). | x | = Absolute Value sign. Ex) | 2 | , | -2 | = 2. 2 units 2 units. -2 -1 0 1 2.

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Solving Absolute Value Equations

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  1. Solving Absolute Value Equations

  2. What is Absolute Value? Absolute Value : The absolute value of a number is the number of units it is from 0 on the number line. (Distance from 0) |x| = Absolute Value sign Ex) |2| , |-2| = 2 2 units 2 units -2 -1 0 1 2

  3. Absolute Value For any real number n: If n≥0, then |n| = n If n≤0, then |n| = -n

  4. Example |7| |7| = 7 3. |0| |0| = 0 -12 is less than 0, so |12| equals –(-12) 2. |-12| |-12| = -(-12) or 12

  5. Absolute Value The absolute value bars can be grouping symbols. Evaluate |4x-5|+ 4.1 if x=-3 |4x-5|+4.1 = |4(-3)-5|+4.1 = |-12-5|+4.1 = |-17|+4.1 = 17+4.1 = 21.1 The value is 21.1 Replace x with -3 Simplify within absolute value bars first Absolute value of -17 is 17

  6. Absolute Value <Rules for absolute value equations> |x|=N • For N: • If N is greater than 0 (N>0), then there are two solutions • If N is equal to 0 (N=0), then there is one solution • If N is less than 0 (N<0), then there is no solution ! Before solving for solutions, you should only leave absolute value at one side. Ex) |x-14|+5=20 |x-14|=15

  7. Example 1. If N is greater than 0 (N>0), then there are two solutions Ex) |x-25|=17 Means x-25=17 or x-25=-17 X-25=-17 X-25+25=-17+25 X=8 X-25=17 X-25+25=17+25 X=42 Check |x-25|=17 |42-25|=17 |17|=17 17=17 Check |x-25|=17 |8-25|=17 |-17|=17 17=17 ✔ ✔

  8. Example 2. If N is equal to 0 (N=0), then there is one solution 0 isn’t negative nor positive, so there is only one solution, 0. Ex) |x+15|=0 Check |x+15|=0 |-15+15|=0 |0|=0 0=0 X+15=0 X=-15 ✔ Ex2) |x-6|+4=4 |x-6|=0 X-6=0 X=6 Check |x-6|+4=4 |6-6|+4=4 |0|+4=4 0+4=4 4=4 ✔

  9. Example 3. If N is less than 0 (N<0), then there is no solution Ex) |2x+7|+5=0 |2x+7|=-5 No solution; absolute value of a number is always positive or zero The solution set that has no members is called empty set (∅)

  10. Exception For equations like |x+4|=2x-2, this rule might not work • For N: • If N is greater than 0 (N>0), then there are two solutions • If N is equal to 0 (N=0), then there is one solution • If N is less than 0 (N<0), then there is no solution

  11. Exception Ex) |x-2|=2x-10 Means x-2=2x-10 or x-2=-(2x-10) X-2=-(2x-10) X-2=-2x+10 x-2+2x=-2x+10+2x 3x-2=10 3x=12 X=4 X-2=2x-10 x-2-x=2x-10-x -2=x-10 8=x Check |x-2|=2x-10 |8-2|=2(8)-10 |6|=16-10 6=6 Check |x-2|=2x-10 |4-2|=2(4)-10 |2|=8-10 2≠-2 ✔

  12. Word Problem Q: For hydrogen to be a liquid, its temperature must be within 2 degrees C of -257 degrees C. It will give positive number and negative number Let b=temperature of hydrogen to be a liquid, then |b-257|=2 Solve it like normal absolute value equations that have two solutions |b-257|=2 Range between possible answers and the number given. b-257=2 b=259 b-257=-2 b=255 The solutions are 259 and 255. Temperature of hydrogen must be in between 255 degrees C and 259 degrees C to be a liquid

  13. Word Problem Q: Hypothermia and hyperthermia are similar words but have opposite meanings. Hypothermia is defined as a lowered body temperature. Hyperthermia means an extremely high body temperature. Both are potentially dangerous conditions and occur when a person’s body temperature is more than 8 degrees above or below the normal body temperature of 98.6 degrees F. At what temperatures do these conditions begin to occur? Let n=normal body temperature. Then |n-98.6|=8 |n-98.6|=8 n-98.6=-8 n=90.6 n-98.6=8 n=106.6 The solutions are 106.6 and 90.6. Hypothermia occurs when the body temperature is below 90.6 degrees F. Hyperthermia occurs when the body temperature is above 106.6 degrees F.

  14. Word Problem Q: A machine is to fill each of several boxes with 16 ounces of sugar. After the boxes are filled, another machine weighs the boxes. If the box is more than 0.2 ounces above or below the desired weight, the box is rejected. At what weights will the box be rejected? Let x=desired weight, then |x-16|=0.2 |x-16|=0.2 x-16=0.2 x=16.2 x-16=-0.2 x=15.8 The solutions are 16.2 and 15.8. The box will be rejected above 16.2 ounces and below 15.8 ounces.

  15. Word Problem Q: Foust Honda is having a contest to win a new Honda Civic. To win a chance at the car, you must guess the number of keys in a jar within 5 of the actual number. The people who are within this range get to try a key in the ignition of the Civic. Suppose there are 697 keys in the jar. What are the highest and lowest guesses that will win a chance at the car? Let x=number of keys within range of 5, then |x-697|=5 |x-697|=5 x-697=-5 x=692 x-697=5 x=702 The lowest guess is 692 and the highest guess is 702. If people guess numbers between 692 and 702, then they would win a chance at the car.

  16. Word Problem Q: A thermometer comes with a guarantee that the stated temperature differs from the actual temperature by no more than 1.5 degrees Fahrenheit. Write and solve an equation to find the minimum and maximum actual temperatures when the thermometer states the temperature is 87.4 degrees Fahrenheit. Let d=actual temperature ±1.5, then |d-87.4|=1.5 |d-87.4|=1.5 d-87.4=1.5 d=88.9 d-87.4=-1.5 d=85.9 The actual temperature is between 85.9 degrees F and 88.9 degrees F.

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