Solving Inequalities

Type 1: Listing values

x is an integer such that -1\leq x \lt 4. List all numbers that satisfy this inequality.

For such questions you need consider if the inequalities are inclusive or strict, in this case we have,

x takes any value greater then or equal to -1       and        x takes any value less than 4

Hence, the integers that satisfy the inequality are: -1,0,1,2,3

Type 2: Solving Inequalities Basic

Solve the inequality 5a - 4 > 2a + 8

Firstly, add 4 to both sides of the inequality to get,

\begin{aligned}(\textcolor{maroon}{+4})\,\,\,\,\,\,\,\,\, 5a -4 &\gt 2a+8 \\ 5a &\gt 2a+12 \end{aligned}

Then, subtract 2a from both sides to get,

\begin{aligned}(\textcolor{maroon}{-2a})\,\,\,\,\,\,\,\,\, 5a &\gt 2a+12  \\ 3a &\gt 12 \end{aligned}

Finally, divide both sides by 3 to get,

\begin{aligned}(\textcolor{maroon}{\div 3})\,\,\,\,\,\,\,\,\, 3a &\gt 12 \\ a &\gt 4 \end{aligned}

Type 3: Solving Inequalities 2 signs

Solve the inequality 5 \lt 2x-3 \lt  13

Firstly, add 3 to each side of the inequality,

(Remember what you do to one side you do to all sides,  even if there are 3 sides), to get

\begin{aligned}(\textcolor{maroon}{+3})\,\,\,\,\,\,\,\,\, 5& \lt 2x-3 \lt 13 \\ 8  &\lt 2x \lt 16 \end{aligned}

Finally, divide both sides by 2 to get,

\begin{aligned}(\textcolor{maroon}{\div 2})\,\,\,\,\,\,\,\,\, 8 \lt 2x&  \lt 16  \\ 4 \lt x& \lt 8 \end{aligned}

Type 4: Multiplying and Dividing by a Negative Number 

When rearranging an inequality, you are performing the same operation to both sides of the inequality without changing it (just like as you would with an equation) but with one exception:

If you multiply or divide by a negative number, then the inequality sign changes direction

For example, if we have to solve the inequality -2x \gt 4, we have to divide both sides by -2,

\begin{aligned}(\textcolor{maroon}{\div -2})\,\,\,\,\,\,\,\,\, -2x &\gt 4 \\ x &\lt -2 \end{aligned}