Linear Inequalities

 
7.1  Inequalities
 
Definition

The relationship between two quantities that do not have the same value.

 
Symbol Meaning
\(\gt\) Greater than
\(\lt\) Less than
 
Example

From the number line above, \(–2\) lies to the left of \(3\).

So, \(–2\) is less than \(3\).

Thus, the inequality is \(–2 < 3\).

 
Example

From the number line above, \(–2\) lies to the right of \(-7\).

So, \(–2\) is greater than \(-7\).

Thus, the inequality is \(–2 > -7\).

 
Describe inequality and form algebraic inequality:
 
Example

From the number line above, \(x\) is less than \(8\).

Thus, \(x\lt 8\).

 
Identify relationship:
 
Symbol Meaning
\(\geq\) Greater than or equal to
\(\leq\) Less than or equal to
 
Properties of inequalities:
 

Converse property of inequality

  •  If \(a \lt b\), then \(b \gt a\).

 
Example

State the converse property of inequality of \(-23\gt-32\).

Answer: \(-32\lt-23\)

 

Transitive property of inequality

  •  If \(a \lt b \lt c\), then \(a \lt c\).

 
Example

State the transitive property of inequality of \(-15\lt-8\lt0\).

Answer: \(-15\lt0\)

 
  • The inequality symbol remains unchanged when adding or subtracting a positive or negative number to or from both sides of the inequality.
 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+c&\lt b+c.\\\\ \end{aligned}\)
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a-c&\lt b-c. \\\\\end{aligned}\)
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+(-c)&\lt b+(-c).\\\\ \end{aligned}\)
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a-(-c)&\lt b-(-c).\end{aligned}\)
 
  • The inequality symbol remains unchanged when multiplying or dividing both sides of the inequality by a positive number.
 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times c&\lt b\times c.\\\\ \end{aligned}\)
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{c}&\lt \dfrac{b}{c}.\end{aligned}\)
 
  • The direction of the inequality symbol is reversed when multiplying or dividing both sides of the inequality by a negative number.
 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times (-c)&\gt b\times (-c).\\\\ \end{aligned}\)
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{-c}&\gt \dfrac{b}{-c}.\end{aligned}\)
 

Additive inverse

  • When both sides of the inequality are multiplied by \(-1\), the direction of the inequality symbol is reversed.
 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,-a&\gt -b.\end{aligned}\)
 

Multiplicative inverse

  • When performing reciprocal of both numbers on both sides of the inequality, the direction of the inequality symbol is reversed.
 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{1}{a}&\gt \dfrac{1}{b}.\end{aligned}\)
 
7.2  Linear Inequalities in One Variable
 
Symbol \(\geq\) Symbol \(\leq\)
  • At least
  • Not less than
  • Minimum
  • At most
  • Not more than
  • Maximum
 
Example

Construct the linear inequality:

The price, \(\text{RM}x\), of a double-storey terrace house is \(\text{RM}450\,000\) and above.

The linear inequality is,

\(x\geq 450\,000\).

 
Solve problems involving linear inequalities in one variable:
 
  • Linear inequality in one variable is an unequal relationship between a number and a variable with power of \(1\).
  • Has more than one possible solution.
 
Example

Calculate: \(16-5x\leq -4\)

\(\begin{aligned} 16-5x&\leq -4 \\\\16-5x-16&\leq -4-16 \\\\-5x&\leq -20 \\\\\dfrac{-5x}{-5}&\geq \dfrac{-20}{-5} \\\\x&\geq 4. \end{aligned}\)

 
Solve simultaneous linear inequalities in one variable:
 
Example

Solve:

\(8x+5\geq 5x-13\) and

\(3x-4\gt 9x+20\).

\(\begin{aligned}8x+5&\geq 5x-13 \\\\8x-5x&\geq-13-5 \\\\3x&\geq-18 \\\\x&\geq -6.\\\\ \end{aligned}\)

\(\begin{aligned}3x-4&\gt 9x+20 \\\\3x-9x&\gt20+4 \\\\-6x&\gt24 \\\\x&\lt-4.\\\\ \end{aligned}\)

Thus, the solution is

\(-6\leq x \lt -4\).