Inequalities

Systems of inequalities are a variant of the material covered in the previous unit, systems of linear equations.  Students often take one look at these questions and feel overwhelmed – however, if you keep a few key things in mind, these problems become much simpler.  A quick refresher on inequalities:

Language

The language used when describing an inequality is usually different from that used to describe an equation.  You will often be asked to “solve” an equation.  In contrast, with an inequality you may be asked to “describe all possible values of \(x\)” or to provide “one possible value of \(x\)”.  This difference in language exists because an inequality is describing a range of solutions.

You should also watch for language like “no more than” or “at least”, which also describe inequalities in word problems.

Signs

In an equation you will see an equals sign, but in an inequality you will see a sign that denotes a relationship.

$$>means\;greater\;than\\\geq means\;greater\;than\;or\;equal\;to\\<means\;less\;than\\\leq means\;less\;than\;or\;equal\;to$$

The open end of the inequality should always open to the greater value.  You can also always think back to the elementary school reminder that the alligator always eats the bigger number.

Strategy:

When solving an inequality, you can perform identical steps to how you would solve an equation (i.e. add 3 to both sides, or divide everything by two) with one exception: when you multiply or divide both sides by a negative number, you must reverse the sign.

$$-3y>6x-5$$

When we divide both sides by \(-1\), the sign reverses.

$$3y<-6x+5$$

Compound Inequalities

A compound inequality is an expression that has multiple inequality signs.  For example, \(9<3x+6<24\).  To solve, make sure that whatever operation you are doing, you do to all three components.  And do not forget to reverse the signs if you divide or multiply by a negative number.

$$9<3x+6<24$$

We can divide each part by 3.

$$3<x+2<8$$

Then we could subtract 2, leaving us with:

$$1<x<6$$

Now we can clearly understand the range of all possible x values.

Graphing Inequalities

Inequalities can be represented in one or two dimensions.  In one dimension, inequalities can be shown on a number line.

An open dot on the number line indicates that that exact number is not included in the set, while a filled dot does include the number it sits on.  In the example above, the line segment would be represented by:

$$-1>x≥2$$

In two dimensions, linear inequalities are graphed with boundary lines, displaying the region of the graph which is included in the set of possible solutions.  A dotted line indicates a \(<\;or\;>\) relationship, while a full line indicates a \(\leq\;or\;\geq\) relationship.  When y is less than (y < … ), you should shade under the line.  When y is greater than (y > …), shade over the line.  For example, the graph of \(y<-2x+1\) would look like: