Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: work out |−7| − |3 − 10|.
Answer
0. |−7| = 7 and |3 − 10| = |−7| = 7. - Solve 5x − 4 = 21.
Answer
x = 5. Add 4 to get 5x = 25, then divide by 5. - A length is measured as 12 cm with an error of at most 0.3 cm. What is the least it could be?
Answer
11.7 cm. Subtract the largest error, 0.3, from 12.
Words used in this lesson
- absolute value
- the distance of a number from zero, never negative. For example: |−9| = 9.
- interval
- all the numbers between two ends. For example: from 10 to 12.
- tolerance
- the greatest error that is allowed. For example: ±5 mL on a bottle.
- percentage error
- the error as a percentage of the true value. For example: 2 out of 50 is 4%.
- triangle inequality
- |a + b| is never more than |a| + |b|. For example: |3 + (−5)| ≤ 3 + 5.
- boundary
- the end of an interval, included or not by the sign. For example: the 12 in x ≤ 12.
Within five millilitres
A bottling plant in Nairobi accepts a bottle if its volume v, in mL, satisfies |v − 500| ≤ 5. The sign ≤ and the 500 together describe something very specific: a bottle can be a little more or a little less than 500 mL, but not too far from it.
Predict first: is a bottle of 494 mL accepted? Is one of 505.5 mL? And if a side is measured 5% too large, will the area be 5% too large, or more? Use the figures to decide, then check.
|x − 1| ≤ 3 means −2 ≤ x ≤ 4. The point x = 6 is 5 from 1, which is more than 3, so it is outside.
Show me the steps
- |x − 1| ≤ 3: the centre is 1 and the radius is 3, so the shaded interval runs from −2 to 4. The test point 6 is 5 away, outside.
- Move x to 3: its distance from 1 is 2, which is at most 3, so it is inside.
- Move x to −2: the distance is 3, exactly r, so it is on the boundary and included because the sign is ≤.
- Change a to −3 and keep r = 3: the interval shifts to −6 ≤ x ≤ 0. The expression becomes |x + 3|, the distance from −3.
True side 5, measured 6: the side is 20.0% too large, but the area is 36 against 25, which is 44.0% too large.
Show me the steps
- True side 5, measured 6: the side is 1/5 = 20.0% too large, and the area 36 against 25 is 11/25 = 44.0% too large.
- Make the error −1: measured side 4, area 16 against 25, error 9/25 = 36.0%. A measurement that is too small gives a smaller area error than one the same size too large.
- Move the true side to 10 with error 1: the side error is 10.0% and the area error is 21/100 = 21.0%. Longer sides make the same error smaller in percentage.
- Set the error to 0: both percentages are 0.0%.
|x − a| is the distance from x to a. |x − a| ≤ r means a − r ≤ x ≤ a + r, the numbers within r of a. |x − a| ≥ r means x ≤ a − r or x ≥ a + r, the numbers at least r away.
|A| = |B| means A = B or A = −B, so there are usually two solutions. For any numbers, |a + b| ≤ |a| + |b|, with equality only when a and b have the same sign (or one is zero).
Errors grow through products: a small percentage error in a length gives about twice that in an area and three times in a volume.
Two common mistakes. Reading |x − 6| < 2 as x < 2: the 6 shifts the middle, and the interval is 4 < x < 8. And giving only one solution to |A| = |B|, or one solution to |x − a| = r: there are two numbers at a given distance from a, one on each side.
Worked examples
1. Solve |x − 5| < 2.
- The numbers within 2 of 5: 5 − 2 < x < 5 + 2.
- So 3 < x < 7.
2. Solve |4x + 1| = |2x − 7|.
- Either the insides are equal, 4x + 1 = 2x − 7, or they are opposite, 4x + 1 = −(2x − 7).
- First: 2x = −8, so x = −4. Second: 4x + 1 = −2x + 7, so 6x = 6 and x = 1.
- Check x = 1: |5| = |−5|. The solutions are x = −4 and x = 1.
3. A side is measured 5% too large. Find the percentage error in the area.
- Measured side = 1.05 × true side, so measured area = 1.05² × true area.
- 1.05² = 1.1025, so the area is 10.25% too large: about 10%, twice the side's error.
Your go. The first step is given. Solve |3x − 1| = 8.
- Either 3x − 1 = 8 or 3x − 1 = −8.
- Solve each equation and check both in the original.
Check the answer
3x = 9 gives x = 3, and 3x = −7 gives x = −7/3. Both give |±8| = 8. The solutions are x = 3 and x = −7/3.
With a GDC (optional)
A GDC has an absolute value function, usually called abs( and found in the number or math menu. Type abs(3 − 10) and it shows 7. It is a function with its own brackets, so always close them: abs(3 − 10) − abs(2 − 9) needs both pairs.
To test an equation such as |5x − 2| = 13, enter your two solutions one at a time and check that abs(5x − 2) shows 13 for each. Use the GDC to check, not to find them: everything here can be done by hand.
Check before you go on
1. Which describes every x with |x + 7| < 2?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. How many integers x satisfy |x − 2| ≤ 6?
3. Solve |2x − 3| = 9. What is the larger solution?
Level 2 · Building
4. Find the sum of all the values of x that satisfy |3x − 6| = |x + 2|.
5. A surveyor measures the side of a square plot as 52 m, but its true side is 50 m. Find the percentage error in the area, to 1 decimal place.
6. |a| = 6 and |b| = 4. Which describes the possible values of |a + b|?
Level 3 · Stretch
7. x is measured as 40 with |x − 40| ≤ 0.5, and y is measured as 25 with |y − 25| ≤ 0.3. What is the largest possible value of |(x − y) − 15|?
8. The side of a cube is measured 3% too large. To the nearest whole percentage, how much too large is the volume found from it?
Something to think about
A side is measured with a 2% error, and the area of the square is worked out from it. By roughly what percentage is the area wrong? What about the volume of a cube, and the radius of a circle whose area is worked out? What pattern do you see in the power?
One way to think about it
Squaring (1.02)² = 1.0404, so the area is wrong by about 4%: twice the side's error. For a cube (1.02)³ = 1.0612, about 6%. A circle's area is proportional to r², so a 2% error in the radius is again about 4% in the area. The percentage error is about the power times the percentage error in the length, as long as the error is small.
Exit ticket
9. Which statement is true for all real numbers a and b?