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Inequalities

November 1, 2009 Leave a comment

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 ).

Examples

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

Reciprocals

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}

Quadratic Inequalties

Quadratic inequalities can often cause problems when trying to decide which range of x values are the solutions. As they are quadratics you usually get two values of x and then have to decide whether it is the values of x between the two bounds that are the solutions or the values of x either side.

For example when solving the inequality:
2x2+x-6>0
we get
(2x-3)(x+2)>0

From this you may be tempted to write x<3/2 and x<-2 however this would be incorrect. The solution must be either -2<3/2, or (x<-2 or x>3/2).
This must be the case for any quadratic with two real roots because all the points between the two roots have one sign and these which are outside the roots have another, as can be clearly seen by considering the general graph.

We can determine which of the two solutions to use by either putting in values to see which work or by drawing a sketch, usually drawing a sketch is the best. For the sketch we only need to consider whether the coefficient of x2, in this case 2, is positive or negative so we know which way the graph curves. The y intercept of the graph is irrelevant.

So in our case because 2>=0 the sketch is a “happy” face so the y>0 either side of the roots and therefore our solutions for x are:
x<-2 or x>3/2

By David Woodford