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Section 2.11 : Linear Inequalities

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

\[2 < \frac{1}{6} - \frac{1}{2}x \le 4\]

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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 \(\frac{1}{6}\) from all the parts. This gives,

\[\frac{{11}}{6} < - \frac{1}{2}x \le \frac{{23}}{6}\] Show Step 2

Finally, all we need to do is multiply all three parts by -2 to get,

\[ - \frac{{11}}{3} > x \ge - \frac{{23}}{3}\]

Don’t forget that because we were multiplying everything by a negative number we needed to switch the direction of the inequalities.

So, the inequality form of the solution is \(\require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{23}}{3} \le x < - \frac{{11}}{3}}}\) (we flipped the inequality around to get the smaller number on the left as that is a more “standard” form). The interval notation form of the solution is \(\require{bbox} \bbox[2pt,border:1px solid black]{{\left[ { - \frac{{23}}{3}, - \frac{{11}}{3}} \right)}}\) .

For the interval notation form remember that the smaller number is always on the left (hence the reason for flipping the inequality form above!) and be careful with parenthesis and square brackets. We use parenthesis if we don’t include the number and square brackets if we do include the number.