Topic 5 Inequalities

For two numbers \(a\) and \(b\):

  • \(a<b\) means that \(a\) is less than \(b\)
  • \(a>b\) means that \(a\) is greater than \(b\)
  • \(a\leq b\) means that \(a\) is less than or equal to \(b\)
  • \(a\geq b\) means that \(a\) is greater than or equal to \(b\)

For example: \(1<2\),  \(-10<-5\),  \(-3\leq -3\),  \(-2\geq -4\).

Note: \(a>0\) means the same as “\(a\) is positive”.

If \(x\) is given on the number line as then the following hold: \(-2<x\),  \(-1>x\),  \(x<0\),  \(-5<x\),  \(x<3\).

The notation “\(\{x : x>3\}\)” means “the set of all numbers \(x\) that are greater than \(3\)”.

Similarly, \(\{x: x\leq -1\}\) is the set of numbers that can be represented on the number line as

The notation \(\{x: -2\leq x <1\}\) can be depicted as

The notation \(\{x: x<-1 \text{ or } x\geq2\}\) can be depicted as

Axioms of inequalities

  1. If \(a<b\) then \(a+c<b+c\).
  2. If \(a<b\) then \(a-c<b-c\).
  3. If \(a<b\) and \(c>0\), then \(ac<bc\).
  4. If \(a<b\) and \(c<0\), then \(ac>bc\).

For example: \(2<3\) \(\iff\) \(-2>-3\).

Example 5.1 Solve \(3x+5>17\).

Solution. \(3x+5>17\)
\(\iff\) \(3x>12\)  (subtract \(5\))
\(\iff\) \(x>4\)  (multiply by \(1/3\))

Example 5.2 Determine the set \(\{x: 2x-x^2<3x-12\}\).

Solution. \(2x-x^2<3x-12\)
\(\iff\) \(2x-x^2-3x+12<0\)  (subtract \(3x-12\))
\(\iff\) \(-x^2-x+12<0\)  (rearrange LHS)
\(\iff\) \(x^2+x-12>0\)  (multiply by \(-1\))
\(\iff\) \((x+4)(x-3)>0\)  (factorise LHS)
\(\iff\) \(\big( x+4>0\) and \(x-3>0\big)\) or \(\big(x+4<0\) and \(x-3<0\big)\)
\(\iff\) \(\big( x>-4\) and \(x>3\big)\) or \(\big(x<-4\) and \(x<3\big)\)
\(\iff\) \(x>3\) or \(x<-4\)
Answer: \(\{x: x>3\text{ or }x<-4\}\).

Example 5.3 Solve \(\dfrac{x-1}{x-2}<0\).

Solution. \(\frac{x-1}{x-2}<0\)
\(\iff\) \((x-1)(x-2)<0\)  (multiply by \((x-1)^2\geq 0\))
\(\iff\) \(\big( x<1\) and \(x>2\big)\) or \(\big(x>1\) and \(x<2\big)\)
\(\iff\) \(1<x<2\)

Example 5.4 Solve \(\dfrac{3x-2}{1+x}\leq 1\).

Solution. \(\frac{3x-2}{1+x}\leq1\)
\(\iff\) \(\frac{(3x-2)-(1+x)}{1+x}\leq0\)
\(\iff\) \(\frac{2x-3}{1+x}\leq0\)
\(\iff\) \(\big( x\leq\frac{3}{2}\) and \(x>-1\big)\) or \(\big(x\geq \frac{3}{2}\) and \(x<-1\big)\)
\(\iff\) \(-1<x\leq\frac{3}{2}\)