1 Properties of absolute value (Section P.1 , page 9)
2 Intervals (Section P.1 , page 8)
3 Linear inequalities (Section 1.6 , page 146)
Rescue crews arrive at an airport in Maine to conduct an air search for a missing Cessna with two people aboard. If the rescue crews locate the missing aircraft, they are to mark the global positioning system (GPS) coordinates and notify the mission base. The search planes normally average 110 miles per hour, but weather conditions can affect the average speed by as much as 15 miles per hour (either slower or faster). If a search plane has 30 gallons of fuel and uses 10 gallons of fuel per hour, what is its possible search range (in miles)? In Example 5, we use an inequality involving absolute value to find the plane’s possible search range.
1 Solve equations involving absolute value.
Recall that geometrically, the absolute value of a real number a is the distance from the origin on the number line to the point with coordinate a The definition of absolute value is
A summary of the most useful properties of absolute value can be found on page 11 in Section P.1.
Because the only two numbers on the number line that are exactly 2 units from the origin are 2 and −2,
Note that if a=0,
Solve each equation.
|x+3|=0
|2x−3|−5=8
Let u=x+3.
Now we check the solution in the original equation.
Check:
We have verified that −3
The solution set is {−3}.
To use the rule for solving equations involving absolute value, we must first isolate the absolute value expression to one side of the equation.
|u|=13
We leave it to you to check these solutions.
The solution set is {−5, 8}.
Solve each equation.
|x−2|=0
|6x−3|−8=1
With a little practice, you may find that you can solve these types of equations without actually writing u and the expression it represents. Then you can work with just the expression itself.
Solve: |x−1|=|x+5|
If |u|=|v|,
The only solution of |x−1|=|x+5|
Solve: |x+2|=|x−3|
Solve: |2x−1|=x+5
Letting u=2x−1
Check:
The solution set of the given equation is {−43, 6}.
Solve |3x−4|=2(x−1).
2 Solve inequalities involving absolute value.
The equation |x|=2
This discussion is equally valid when we use any positive real number a instead of the number 2, and it leads to the following rules for replacing inequalities involving absolute value with equivalent inequalities that do not involve absolute value.
Solve the inequality |4x−1|≤9
Rule 2 applies here with u=4x−1
The solution set is {x|−2≤x≤52};
Solve |3x+3|≤6
Note the following techniques for solving inequalities similar to Example 4.
To solve |4x−1|<9,
To solve |1−4x|≤9,
In the introduction to this section, we wanted to find the possible search range (in miles) for a search plane that has 30 gallons of fuel and uses 10 gallons of fuel per hour. We were told that the search plane normally averages 110 miles per hour but that weather conditions can affect the average speed by as much as 15 miles per hour (either slower or faster). How do we find the possible search range?
To find the distance a plane flies (in miles), we need to know how long (time, in hours) it flies and how fast (speed, in miles per hour) it flies.
Let x=actual
Solve this inequality for x.
The actual speed of the search plane is between 95 and 125 miles per hour.
Because the plane uses 10 gallons of fuel per hour and has 30 gallons of fuel, it can fly 3010,
The search plane’s range is between 285 and 375 miles.
Repeat Example 5 , but let the average speed of the plane be 115 miles per hour and suppose the wind speed can affect the average speed by 25 miles per hour.
Solve the inequality |2x−8|≥4
Rule 4 applies here with u=2x−8 and a=4.
The solution set is {x|x≤2 or x ≥6};
Solve |2x+3|≥6
Solve each inequality.
|3x−2|>−5
|5x+3|≤−2
Because the absolute value is always nonnegative, |3x−2|>−5
There is no real number with absolute value ≤−2
Solve.
|5−9x|>−3
|7x−4|≤−1
Solve: |x+1|<3|x−1|
We solve the two inequalities:
From Figure 1.16, we see that both inequalities are true on S1∩S2.
Solve: |x−2|<4|x+4|
You can use the “test point“ method to verify that the solution set of 2x−1x−1>0
In Exercises 1–6, assume that a>0.
The solution set of the equation |x|=a
The solution set of the inequality |x|<a
The solution set of the inequality |x|≥a
The equation |u|=|v|
True or False. If a<b,
True or False. The statement “a is nonnegative” can be expressed as an inequality by “a>0
True or False. The solution set of |3x−2|<a
True or False. If a>0,
In Exercises 9–36, solve each equation.
|3x|=9
|4x|=24
|−2x|=6
|−x|=3
|x+3|=2
|x−4|=1
|6−2x|=8
|6−3x|=9
|6x−2|=9
|6x−3|=9
|2x+3|−1=0
|2x−3|−1=0
12|x|=3
35|x|=6
|14x+2|=3
|32x−1|=3
6|1−2x|−8=10
5|1−4x|+10=15
2|3x−4|+9=7
9|2x−3|+2=−7
|2x+1|=−1
|3x+7|=−2
|x2−4|=0
|9−x2|=0
|1−2x|=3
|4−3x|=5
|13−x|=23
|25−x|=15
In Exercises 37–46, solve each equation.
|x+3|=|x+5|
|x+4|=|x−8|
|3x−2|=|6x+7|
|2x−4|=|4x+6|
|2x−1|=x+1
|3x−4|=2(x−1)
|4−3x|=x−1
|2−3x|=2x−1
|3x+2|=2(x−1)
|4x+7|=x+1
In Exercises 47–62, solve each inequality.
|3x|<12
|2x|≤6
|4x|>16
|3x|>15
|x+1|<3
|x−4|<1
|x|+2≥−1
|x|+2>−7
|2x−3|<4
|4x−6|≤6
|5−2x|>3
|3x−3|≥15
|3x+4|≤19
|9−7x|<23
|2x−15|<0
|x+5|≤−3
In Exercises 63–72, solve each inequality.
|x−2x+3|<1
|x+3x−1|<2
|2x−3x+1|≤3
|2x−13x+2|≤1
|x−1x+2|≥2
|x+3x−2|≥3
|2x+1x−1|>4
|2x−13x+2|>5
|x−1|≤2|2x−5|
2|x−5|≤|2x−3|
Varying temperatures. The inequality |T−75|≤20,
Scale error. A butcher’s scale is accurate to within ±0.05
Company budget. A company budgets $700 for office supplies. The actual expense for budget supplies must be within ±$50
National achievement scores. Suppose 68% of the scores on a national achievement exam will be within ±90
Blood pressure. Suppose 60% of Americans have a systolic blood pressure reading of 120, plus or minus 6.75. Let x=a
Gas mileage. A motorcycle has approximately 4 gallons of gas, with an error margin of ±14
Event planning. An event planner expects about 120 people at a private party. She knows that this estimate could be off by 15 people (more or fewer). Food for the event costs $48 per person. How much might the event planner’s food expense be?
Company bonuses. Bonuses at a company are usually given to about 60 people each year. The bonuses are $1200 each, and the estimate of 60 recipients may be off by 7 people (more or fewer). How much might the company spend on bonuses this year?
Fishing revenue. Sarah sells the fish she catches to a local restaurant for 60 cents a pound. She can usually estimate how much her catch weighs to within ±12
Ticket sales. Ticket sales at an amusement park average about 460 on a Sunday. If actual Sunday sales are never more than 25 above or below the average and if each ticket costs $29.50, how much money might the park take in on a Sunday?
In Exercises 83–88, solve each equation.
|x2−9|=x−3
|x2−8|=−2x
|x2−5x|=6
|x2+3x−2|=2
|x2−7|=|6x|
|x2−2x|=|5x−10|
|2x2−3x+5|=|x2−4x+7|
|x2+x+3|=|x2+5x+1|
|2x−3|+|x−2|=4
[Consider three cases:
−∞ <x≤32,
32<x≤2,
2<x<∞.
2|2x−3|−3|x−2|=5
|x|2−4|x|−7=5
2|x|2−|x|+8=11
Show that if 0<a<b
Show that if 0<a<b<c
Show that if x2<a
Show that if x2>a
In Exercises 93–104, write an absolute value inequality with variable x whose solution set is in the given interval(s).
(1, 7)
[3, 8]
[−2, 10]
(−7, −1)
(−∞, 3)∪(11, ∞)
(−∞, −1)∪(5, ∞)
(−∞, −5]∪[10, ∞)
(−∞, −3]∪[−1, ∞)
(a, b)
[c, d]
(−∞, a)∪(b, ∞)
(−∞, c]∪[d, ∞)
Varying temperatures. The weather forecast predicts that temperatures will be between 8°
In Exercises 106–111, solve each inequality for x.
|x−1|≤5
1≤|x−2|≤3
|x−1|+|x−2|≤4
|x−2|+|x−4|≥8
|x−1|+|x−2|+|x−3|≤6
|x−1| − (x+2)x+2≥0
For which values of x is √(x−3)2=x−3?
For which values of x is
Solve |x−3|2−7|x−3|+10=0.
[Hint: Let u=|x−3|.
In Exercises 115–122, simplify each expression.
2+52
−3+72
−3−72
(a−b)−(a+b)2
√(5−2)2+(3−7)2
√(−8+3)2+(−5−7)2
√(2−5)2+(8−6)2
√(√3−√12)2+(√2+√8)2
In Exercises 123–128, add the appropriate term to each binomial so that it becomes a perfect square trinomial. Write and factor the trinomial.
x2+4x
x2−6x
x2−5x
x2+7x
x2+32x
x2−45x