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Linear Inequalities

Linear inequalities

Inequalities are mathematical statements similar to equations, but with the equals sign replaced with one of the following four inequality symbols

Symbol Meaning
$\lt$ less than: $a\lt b$ - the value of $a$ is strictly less than the value of $b$
$\gt$  greater than: $a\gt b$ - the value of $a$ is strictly greater than the value of $b$
$\leq$ less than or equal to : $a\leq b$ - the value of $a$ is either less than the value of $b$ or possibly equal to the value of $b$
$\geq$ greater than or equal to : $a\geq b$ - the value of $a$ is either greater than the value of $b$ or possibly equal to the value of $b$

The method of solving an inequality is effectively identical to the process of solving an equation; but with a couple of caveats

  • multiplication or division by a negative number changes the direction of an inequality ($\gt$ becomes $\lt$ and vice versa)
  • The solution is still an inequality

Linear inequalities 1

Solve the following inequality:

$$ 3x - 10 \lt 23 $$
solution - press button to display
$$ \begin{align} 3x - 10 &\lt 23 \\ 3x &\lt 33 \\x &\lt 11 \end{align} $$

Linear inequalities 2

Solve the following inequality $$ 4 - 2t \leq 20 $$
solution - press button to display

Note that the coefficient of the variable is negative so some care needs to be taken

Approach 1

Avoiding multiplication and division by negative numbers:

$$\begin{align}4 - 2t &\leq 20\\ 4 &\leq 20 + 2t \\ -16 &\leq 2t \\ -8 &\leq t\end{align}$$

Approach 2 

Multiplication or division by a negative number changes the direction of the sign

$$\begin{align}4-2t &\leq 20 \\ -2t &\leq 16 \\ t &\geq -8 \end{align}$$

 

Justification of the change of orientation of the sign requires a little thought, but we see both methods return an equivalent result. 

 

Linear inequalities 3

Solve the following system of inequalities:

$$ \begin{align} 4x + 6 &\gt 22 \\ 40-2x &\geq 10 \end{align} $$

In addition, state the cardinality (number of members) of the set of integers that satisfy the inequalities.

solution - press button to display

The first inequality reduces to $$ \begin{align} 4x + 6&\gt 22 \\ 4x &\gt 16 \\ x &\gt 4 \end{align} $$

The second yields $$ \begin{align} 40 - 2x &\geq 10\\ -2x &\geq -30 \\x &\leq 15 \end{align} $$

Hence the solution of the set of inequalities is $$4\lt x\leq 15 $$

The set of integer solutions is $\left\{5,6,7,8,9,10,11,12,13,14,15\right\}$ which has a cardinality (number of elements) of 11