Algebra · Junior Cert
Inequalities
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Section 1 of 5
Reading Inequalities
An inequality is a statement about size. You already know loads of them from real life:
Age $16$ or more — drive a tractor
$x \ge 17$ — drive a car
$x \ge 18$ — vote
$x \le 65$ — work
$0 \le x \le 90$ — length of a soccer game
Read these
$x > 5$ — $x$ is greater than $5$
$x < 20$ — $x$ is less than $20$
$19 < x < 90$ — $x$ is greater than $19$ and less than $90$
$x \ge 16$ — $x$ is $16$ or more (greater than or equal to $16$)
Section 2 of 5
Number Systems
Which numbers are allowed? That is set by the number system.
Natural numbers $\mathbb{N}$ — whole, positive: $\{1,2,3,\dots\}$
Integers $\mathbb{Z}$ — whole, positive and negative: $\{\dots,-3,-2,-1,0,1,2,3,\dots\}$
Rational $\mathbb{Q}$ — any number that can be written as a fraction
Real $\mathbb{R}$ — any number that exists (rational and irrational together)
Section 3 of 5
Number Lines
Show $x \le 3$ on a number line — it looks different in each number system.
(i) $x \in \mathbb{N}$ — just the naturals up to $3$:
(ii) $x \in \mathbb{Z}$ — integers, continuing left:
(iii) $x \in \mathbb{R}$, $x \le 3$ — solid line, filled dot (included):
(iv) $x \in \mathbb{R}$, $x < 3$ — open dot (not included):
(v) $x \in \mathbb{R}$, $-1 \le x < 2$ — a segment, filled left, open right:
Section 4 of 5
Solving Inequalities
Solve just like an equation — but flip the sign whenever you multiply or divide by a negative. (Think: $3>2$ but $-3<-2$.)
Worked example
Solve $2x-1 \le 11$.
$2x \le 12$
$x \le 6$
Worked example
Solve $3x+2 \ge x+10$.
$3x-x \ge 10-2$
$2x \ge 8$
$x \ge 4$
Worked example — flip the sign
Solve $1-2x \ge 9$.
$-2x \ge 8$
Divide by $-2$ and flip: $2x \le -8$
$x \le -4$
You try
Solve $1-3x \le 22$.
Pen and paper out — try it before you reveal.
$-3x \le 21$
Divide by $-3$ and flip: $3x \ge -21$
$x \ge -7$
$x \ge -7$
Section 5 of 5
Solve & Show on a Line
Worked example
Solve $3x-1 \le 5$, $x \in \mathbb{N}$, and show it.
$3x \le 6$
$x \le 2$
Naturals: $\{1,2\}$
You try
Solve $1-2x \le -3$, $x \in \mathbb{Z}$, and show it.
Pen and paper out — try it before you reveal.
$-2x \le -3-1$
$-2x \le -4$
Divide by $-2$ and flip: $2x \ge 4 \Rightarrow x \ge 2$
$x \ge 2$, i.e. $\{2,3,4,\dots\}$
$x \ge 2$, i.e. $\{2,3,4,\dots\}$
That’s inequalities.
Read them, know your number sets, solve like an equation — and flip on a negative.