Inequalities and the Number Line Slideshow 51, Mathematics Mr Richard Sasaki, Room 307

Objectives Recall the number line Understand how values on the number line and inequalities relate

The Number Line The real number line contains all non- complex numbers from −∞ to ∞. Numbers should have equal distance so we can only show a part of the number line. We usually use zero at the centre. Numbers get less as you go further left and get more as you go further right.

The opposite of inequality is equality. Equality has the symbol… Inequalities The opposite of inequality is equality. Equality has the symbol… = There are four types of inequality. < Less than > Greater than / More than ≤ Less than or equal to ≥ Greater than or equal to / More than or equal to

Inequalities To make sure that your inequality is the right way around, make sure that the smaller number is at the tip / point and the larger number is at the mouth. 6 < 9 The symbol ‘≤’ shouldn’t be used because it is clear that 6 ≠ 9.

< 𝑥 6 Inequalities If we have a statement… This means that 𝑥 must be less than 6 but can’t be 6. Do you know how to show this on the number line? The hollow circle implies that it covers everything below but not including 6.

< 𝑥 6 Inequalities Does 𝑥<6 include 5.9? Yes! So, we certainly can’t say something like 𝑥<6≡𝑥≤5.9

≤ 𝑥 6 Inequalities If we have a statement… This means that 𝑥 must be less than 6 and can be 6. Do you know how to show this on the number line? The filled circle implies that it covers everything below and including 6.

Answers (Part 1) 𝑥≤3 𝑥>−2 𝑥≥1 1, 2, 3, 4, 5 −1, 0, 1, 2, 3 𝑥≥−2 𝑥<1 𝑥>−4 −2, −1, 0, 1, 2 −3, −2, −1, 0, 1 −4, −3, −2, −1, 0 3, 4, 5, 6, 7 −6, −5, −4, −3, −2, 𝑥≤0 𝑥>2 𝑥≤−1 −4, −3, −2, −1, 0 −5, −4, −3, −2, −1 3, 4, 5, 6, 7 −2, −1, 0, 1, 2 2, 3, 4, 5, 6

Inequalities 𝑥 ≤ 6 We know that this includes 6 but certainly doesn’t contain 6.00001 or something. So far we have only spoken about real numbers where 𝑥∈ℝ. Inequalities have different meanings for integers where 𝑥∈ℤ.

Inequalities If 𝑥∈ℤ, can we say… 𝑥<6≡𝑥≤5 Yes! 𝑥≥6≡𝑥>5 Yes! Remember, probabilities themselves are real numbers where 0≤𝑥≤1, 𝑥∈ℝ. This is not true about values on dice (for example) however! On a die… 1≤𝑥≤6, 𝑥∈ℤ How could we write this with less than symbols? 0<𝑥<7, 𝑥∈ℤ

One Last thing… When we have inequalities that bound an unknown in both directions they must both say less than (or more than but this is a bit strange). How would we say this relationship? 1<𝑥≤6 Both symbols must point the way, this makes no sense for one single unknown! 1<𝑥≥6

Answers – Part 2 0, 1, 2, 3, 4 1≤𝑥<6 3<𝑥≤7 2<𝑥<5 −4, −3, −2, −1, 0 −5, −4, −3, −2, −1 2<𝑥<5 4, 5, 6, 7 1, 2, 3, 4, 5 3, 4 −5, −4, −3, −2, −1 3, 4, 5, 6, 7 −2≤𝑥≤1 −1≤𝑥<3 −6, −5, −4, −3, −2 −3<𝑥<3 −2, −1, 0, 1 −1, 0, 1, 2 −2, −1, 0, 1, 2 1, 2, 3, 4 6 4, 5, 6, 7 0, 1, 2, 3, 4