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An inequality(or inequation) is similar to an equation accept for instead of saying both side of the inequality are equal we say one side is greater than (or equal to depending upon the type of inequality) the other, this is done using the greater than (> ), greater than or equal to (\geq ), less than (< ) and less than or equal to (\leq ).


Some simple examples which contain only one variable are:

5x > 3
2x-7 < x+5
x^2 - 4 \leq x

Solving and Manipulating Inequalities

Inequalities can be solved by rearranging them and isolating the variable you want to find in a similar way to normal equation (see the post quadratic inequalities to see how to solve quadratics). However, rather than getting an exact value such as x=3 we get a range (open or closed) of values such as x<2 or -3<-1.

Much of the manipulation is the same though there are slight variations when dividing or multiplying by negative numbers or taking the reciprocal. The important thing to remember is that like normal equations we must do the same to both sides.

Addition and Subtraction

Addition and subtraction are exactly the same to equalities. We can add or subtract whatever we like as long as we do the same to both the sides. This enables us to take expressions “to the other side” by reversing their sign. For example all the following manipulations are valid.

x + 3 < 4  \Leftrightarrow x < 4-3 = 1
x - 3 < 4  \Leftrightarrow x < 4+3 = 7
x < 4  \Leftrightarrow x + 3 < 4+3 = 7

Multiplication and Division

Again we can perform multiplication and division in a similar way to the way we perform it with equalities by doing the same to both sides. However, if we are multiplying or dividing by a negative number we must reverse the direction of the inequality since
-x < y \Leftrightarrow x>-y
This means we must be careful when diving by an unknown since by definition we don’t know whether or not it is positive or negative. If this has to be done you should consider both the cases it is positive and negative separately and if it is only positive or negative then the other inequality should lead to a contradiction which can easily be spotted such as x<0 and x>3.

Examples of valid manipulation are below:
2x < 6 \Leftrightarrow x<3
-2x < 6 \Leftrightarrow x>3
\frac{4}{x} < 3 \Leftrightarrow \frac{4}{3} < x for x \geq 0 and/or \frac{4}{3} > x for x<0


When taking the reciprocal or “one over” of an expression you must reverse the inequality so
x < y \Leftrightarrow \frac{1}{x} > \frac{1}{y}

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