Any equations that DO NOT use equal sign (=) involve inequalities.

\(\enspace \neq \quad \text{not equal to}\)

\(\enspace > \quad \text{greater than}\)

\(\enspace \geq \quad \text{greater than or equal to}\)

\(\enspace < \quad \text{less than}\)

\(\enspace \leq \quad \text{less than or equal to}\)


Negative Number

 Dividing or multiplying a linear inequality with a negative number will change the sign of inequality!  

Example Divide the inequality \(5 > 2\) with -1.

 Solution 
\( 5 > 2 \\ \, \\ \dfrac{5}{-1} {\color{red}{<}} \dfrac{2}{-1} \\ \, \\ -5 < -2 \\ \, \\ \)

Example Multiply the inequality \(3 > 1\) with -3.

 Solution 
\( 3 > 1 \\ \, \\ (3)\cdot({\color{red}{-3}}) \, {\color{red}{<}} \, (1)\cdot({\color{red}{-3}}) \\ \, \\ -9 < -1 \\ \, \\ \)

Example Solve the inequality \(-7x+7 > 0\) for \( x \).

 Solution 
\( -7x+7 > 0 \\ \, \\ -7x+7 {\color{red}{-7}} > 0 {\color{red}{-7}} \\ \, \\ -7x > -7 \\ \, \\ \dfrac{7x}{{\color{red}{-7}}} {\color{red}{<}} \dfrac{-7}{{\color{red}{-7}}} \\ \, \\ x < 1 \\ \, \\ \)

Solve the inequality \(2x-8 > 4\) for \(x\).
Answer
\( 2x-8 {\color{red}{+8}} > 4 {\color{red}{+8}} \\ \, \\ 2x > 12 \\ \, \\ \dfrac{2x}{{\color{red}{2}}} > \dfrac{12}{{\color{red}{2}}} \\ \, \\ x > 6 \)


Solve the inequality \(\dfrac{3x-1}{-5} < 8\) for \(x\).
Answer
\( \dfrac{3x-1}{-5}\cdot{\color{red}{-5}} \enspace {\color{red}{>}} \enspace 8\cdot{\color{red}{-5}} \\ \, \\ 3x-1 > -40 \\ \, \\ 3x-1 {\color{red}{+1}} > -40{\color{red}{+1}} \\ \, \\ 3x > -39 \\ \, \\ \dfrac{3x}{{\color{red}{3}}} > \dfrac{-39}{{\color{red}{3}}} \\ \, \\ x > -13 \)