Absolute value and inequalities on the number lineIB MYP Maths Extended: Subtopic test

1. 1
   Two points on a number line are at $x=-7$ and $x=3$.

   (a)

   Find the value of $|-7|-|3|$.

   [1 mark]

   • A$4$
   • B$-4$
   • C$10$
   • D$-10$

   (b)

   What is the distance between the two points?

   [1 mark]

   • A$4$
   • B$10$
   • C$-10$
   • D$21$

   (c)

   A third point is exactly $5$ units from the point $x=-7$. Find the two possible positions of the third point.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A laboratory freezer must keep its temperature $T$ (in $^\circ$C) within the range $-25\le T<-18$.

   (a)

   Which of these temperatures is allowed?

   [1 mark]

   • A$-10^\circ$C
   • B$-18^\circ$C
   • C$-30^\circ$C
   • D$-25^\circ$C

   (b)

   Which is the correct interval notation for the allowed temperatures?

   [1 mark]

   • A$(-25,-18]$
   • B$[-25,-18]$
   • C$[-25,-18)$
   • D$(-25,-18)$

   (c)

   List all the whole-number temperatures that are allowed.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The water temperature $t$ (in $^\circ$C) in a school swimming pool is acceptable if $|t-27|\le 2$.

   (a)

   Write the acceptable temperatures (i) as a double inequality and (ii) in interval notation.

   [3 marks]

   (b)

   On Monday the temperature is $24.5^\circ$C. Later a heater raises it by $1.8^\circ$C. By working out $|t-27|$ each time, decide whether the temperature is acceptable (i) on Monday and (ii) after the heater has been on.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory makes metal rods with a target length of $50$ cm. A rod passes inspection if its length $L$ cm satisfies $|L-50|\le d$, where $d$ cm is the tolerance.

   (a)

   (i) Write the interval of passing lengths in interval notation for $d=1$, $d=2$ and $d=3$.
   (ii) Describe the pattern and write a general rule for the interval in terms of $d$.
   (iii) Use your rule to give the interval when $d=0.4$, and verify that a rod of length $50.5$ cm fails.

   [6 marks]

   (b)

   The tolerance is $d=0.4$. An inspector's ruler measures to the nearest $0.5$ cm and records a rod as $50.5$ cm.
   (i) Write an inequality for the actual length $L$ of a rod recorded as $50.5$ cm.
   (ii) Justify why the inspector cannot be sure that this rod passes.
   (iii) A better ruler measures to the nearest $0.1$ cm and also records $50.5$ cm. Show that this rod fails.

   [6 marks]

   Total for question 4: 12 marks

End of questions