Solutions for Lesson 8 Practice Questions

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Contents

Intervals

Write the regions marked on the lines in interval notation.

part a)

(-\infty, 3)

part b)

[0,3] \union (6, \infty)

Absolute Value

Give the value of each of the following without the absolute value signs.

a) |9|

|9| = 9

b) |-9|

|-9| = 9

c) |-3+2|

|-3+2| = 1

d) |3-2|

|3-2| = 1

e) |-3-2|

|-3-2| = 5

Distance on the Number Line

Write an expression to show that the distance...

a) between x and 4 on a number line is equal to 9.

|x-4| = 9 or |4-x| = 9

b) between x and -5 on a number line is equal to 7.

|x+5| = 7 or |-5-x| = 7

Strict Inequality

If x < 5 , then...

a) we can equivalently say x can equal any number up to and including 4.\overline 9

false

b) we can equivalently say x can equal any number up to and including 5.

false

c) we can equivalently say x can equal any number up to and including 4.9.

false

d) we can’t give a maximum value that x can equal because whatever number we choose, we can always find a number larger than that that is still smaller than 5.

true

Prove Me Wrong

Suppose I claim that x < 10 means that x can be any number up to and including 9

a) Prove me wrong by providing an example of a number which satisfies the inequality but is closer to 10 than to 9

Your answer can be any number that is greater than 9 but less than 10.  Example: 9.5

b) So now I claim that your answer is the largest that x can be.  Prove me wrong again by providing another number closer to 10 than your number is.

Your answer can be any number that is less than 10 but greater than the answer you put for part a).  Since I put 9.5 in part a), I could put something like 9.7 here.

c) If we carry on, will I eventually find the largest number possible that is <10, or will you always be able to prove me wrong? If you think that you will always be able to prove me wrong, type prove in the box, and if not, put largest

prove

And or Or

Mark the regions on the number lines that satisfy the inequalities given.

a) x>3 and x<5

b) x>3 or x<5

Drug Variation 1

According to the site Science Based Medicine, “Most regulators worldwide have decided that a 20% variation [in the active ingredient of a tablet] is generally not clinically significant.”

Acetylsalicylic acid is a painkiller originally derived from the bark of a willow tree.  It’s often sold under the trade name ‘aspirin’.  In the USA, a tablet of this drug usually contains 325 mg of active ingredient, that is, of acetylsalicylic acid rather than any additional ingredients.  In the UK, a tablet usually contains 300 mg.

According to the information above, by how many milligrams (mg) can the amount of active ingredient vary...

a) in the USA tablet?

65 mg

b) in the UK tablet?

60 mg

Drug Variation 2

According to the site Science Based Medicine, “Most regulators worldwide have decided that a 20% variation [in the active ingredient of a tablet] is generally not clinically significant.”

Acetylsalicylic acid is a painkiller originally derived from the bark of a willow tree.  It’s often sold under the trade name ‘aspirin’.  In the USA, a tablet of this drug usually contains 325 mg of active ingredient, that is, of acetylsalicylic acid rather than any additional ingredients.  In the UK, a tablet usually contains 300 mg.

Sometimes, the variation is written as the standard amount \pm amount it’s allowed to vary, where \pm means plus or minus.  How would you write this for...

c) USA

325 \pm 65

d) UK

300 \pm 60

Drug Variation 3

According to the site Science Based Medicine, “Most regulators worldwide have decided that a 20% variation [in the active ingredient of a tablet] is generally not clinically significant.”

Acetylsalicylic acid is a painkiller originally derived from the bark of a willow tree.  It’s often sold under the trade name ‘aspirin’.  In the USA, a tablet of this drug usually contains 325 mg of active ingredient, that is, of acetylsalicylic acid rather than any additional ingredients.  In the UK, a tablet usually contains 300 mg.

Suppose you have a tablet.  It contains x milligrams of active ingredient.  How can you write that the difference between x and the standard amount for each tablet is at most the variation?

e) USA

|x-325| \le 65

f) UK

|x-300| \le 6

Drug Variation 4

According to the site Science Based Medicine, “Most regulators worldwide have decided that a 20% variation [in the active ingredient of a tablet] is generally not clinically significant.”

Acetylsalicylic acid is a painkiller originally derived from the bark of a willow tree.  It’s often sold under the trade name ‘aspirin’.  In the USA, a tablet of this drug usually contains 325 mg of active ingredient, that is, of acetylsalicylic acid rather than any additional ingredients.  In the UK, a tablet usually contains 300 mg.

USA:

|x-325| \le 65

UK:

|x-30| \le 60

Either by solving the inequalities in the previous question, or by thinking about the question, write the interval of values for x using inequalities.

e) USA
|x-325| \le 65

x-325 \le 65 and 325-x \ge 65
x\le 65 + 325 and -x\ge 65-325
x\le 390 and x\ge 260
260\le x\le 390

f) UK
|x-300| \le 60

x-300 \le 60 and 300-x \ge 60
x\le 60 + 300 and -x\ge 60-300
x\le 360 and x\ge 240
240\le x\le 360

Challenge

Considering the conditions given, write the following without the absolute value signs.  For example, if x < -5, then x-1 is negative, so |x-1| = 1-x

a) |x-5| where x>7

x-5

b) |x+2| where x<-3

-x-2

c) |2-x| where x>2

x-2