Paul's Online Notes
Paul's Online Notes
Home / Algebra / Solving Equations and Inequalities / Linear Inequalities
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 2.11 : Linear Inequalities

3. Solve the following inequality and give the solution in both inequality and interval notation.

\[ - 1 < 4x + 2 < 10\]

Show All Steps Hide All Steps

Hint : Solving double inequalities uses the same basic process as solving single inequalities. Just remember that what you do to one part you have to do to all parts of the inequality.
Start Solution

Just like with single inequalities solving these follow pretty much the same process as solving a linear equation. The only difference between this and a single inequality is that we now have three parts of the inequality and so we just need to remember that what we do to one part we need to do to all parts.

Also, recall that the main goal is to get the variable all by itself in the middle and all the numbers on the two outer parts of the inequality.

So, let’s start by subtracting 2 from all the parts. This gives,

\[ - 3 < 4x < 8\] Show Step 2

Finally, all we need to do is divide all three parts by 4 to get,

\[ - \frac{3}{4} < x < 2\]

So, the inequality form of the solution is \(\require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{3}{4} < x < 2}}\) and the interval notation form of the solution is \(\require{bbox} \bbox[2pt,border:1px solid black]{{\left( { - \frac{3}{4},2} \right)}}\) .