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Double Inequalities,
How to Solve Inequalities


The inequalities introduction along with the solving inequalities page showed how to write inequalities in basic form, and how to solve such inequalities.

We can also learn how to write inequalities as what are known as “double inequalities”, which is when there are two inequalities symbols present instead of just one.


So in essence we have to inequalities in one, when dealing with a double inequality.




How to Solve Double Inequalities:


Looking at numbers, we can write that  5  is less than  6,  and  6  is less than  7  as a double inequality.

5 < 6 < 7


It is a bit neater and easier than writing out two individual inequalities.

5 < 6 \space \space , \space \space 6 < 7


Just like with individual inequalities, double inequalities can be solved also.






Solving Double Inequalities


We could be asked to solve something like,   {\text{-}}4 < 2x < 6.

What we want to do is get the variable x by itself,
like in the case of single inequalities.


This can be done here by dividing through the double inequality by 2, each part needs to have the operation performed.

\frac{{\text{-}}4}{2} \space {\scriptsize{<}} \space \frac{2x}{2} \space {\scriptsize{<}} \space \frac{6}{2}    =    {\text{-}}2 < x < 3


x values less than 3 and greater than 2 are solutions to the inequality.




Examples    



(1.1) 

Solve    1 < x + 3 < 5.

Solution   

1 \space {\text{–}} \space 3 \space < \space x + 3 \space {\text{--}} \space 3 \space < \space 5 \space {\text{--}} \space 3

{\text{-}}3 \space {\footnotesize{<}} \space x \space {\footnotesize{<}} \space 2

x values less than 2 and greater than –3 are solutions to the inequality.

Graph:
A graph that may be required to be drawn when learning how to solve double inequalities.




(1.2) 

Solve    10 \space {\footnotesize{\le}} \space 4x + 6 \space {\footnotesize{<}} \space 18.

Solution   

10 \space {\text{–}} \space 6 \space \le \space 4x + 6 \space {\text{–}} \space 6 \space < \space 18 \space {\text{--}} \space 6

4 \space \le \space 4x \space < \space 12

\frac{4}{4} \space \le \space \frac{4x}{4} \space < \space \frac{12}{4}

1 \space \le \space x \space < \space 3

x values less than 3 and greater than or equal to 1 are solutions to the inequality.

Graph:
Double inequality graph where one of the values is included in the solution.




(1.3) 

Solve    {\text{-}}3 \space {\footnotesize{\le}} \space 5 \space {\text{–}} \space 2x \space {\footnotesize{<}} \space 11.

Solution   

{\text{-}}3 \space {\text{–}} \space 5 \space {\footnotesize{\le}} \space 5 \space {\text{–}} \space 2x \space {\text{–}} \space 5 \space {\footnotesize{<}} \space 11 \space {\text{--}} \space 5

{\text{-}}8 \space {\footnotesize{\le}} \space {\text{-}}2x \space {\footnotesize{<}} \space 6

\frac{{\text{-}}8}{{\text{-}}2} \space \le \space \frac{{\text{-}}2x}{{\text{-}}2} \space < \space \frac{6}{{\text{-}}2}

Dividing by a negative number flips the inequality signs.

4 \space \ge \space x \space > \space {\text{-}}3

x values less than or equal to 4 and greater than –3 are solutions to the inequality.

Graph:
Inequality graph where the solution range takes us into the negative number range.





(1.4) 

Solve    4 \space {\footnotesize{<}} \space 2 \space {\text{--}} \space a \space {\footnotesize{\le}} \space 8.

Solution   

4 \space {\text{–}} \space 2 \space {\footnotesize{<}} \space 2 \space {\text{--}} \space x \space {\text{--}} \space 2 \space {\footnotesize{<}} \space 8 \space {\text{--}} \space 2

2 \space < \space {\text{-}}x \space \le \space 6

2 {\footnotesize{\times}} ({\text{-}}1) \space < \space {\text{-}}x {\footnotesize{\times}} ({\text{-}}1) \space \le \space 6 {\footnotesize{\times}} ({\text{-}}1)

Like division, multiplying by a negative number flips the inequality signs.

{\text{-}}2 \space > \space x \space \ge \space {\text{-}}6

x values less than –2 and greater than or equal to –6 are solutions to the inequality.

Graph:
Graph where both values in the solution take us into the negative range of numbers.






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