Revision:OCR Core 1 - Inequalities

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Revision:OCR Core 1 - Inequalities

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 1 - Inequalities

1 6. Inequalities
2 Also See
3 Comments

6. Inequalities

Notation for inequalities

In some cases one will be given an inequality; a mathematical statement that shows how something is greater than, less than, greater than or equal to, less than or equal to, etc. another quantity. When one is asked to solve these, one is aiming to give an expression for the set of numbers for which the unknown will fit the expression.

The following meanings are meant for a quick reminder, and students who do not know them should certainly review the topic at a lower level before proceeding.

x > y

This means that "x" is greater than "y".

x < y

This means that "x" is less than "y".

x >= y

This means that "x" is greater than or equal to "y".

x <= y

This means that "x" is less than or equal to "y".

The less than, or greater than inequality operands are called strict inequalities, and those involving less than or equal to, or greater than or equal to are called weak inequalities.

Solving linear inequalities

The process of solving linear inequalities is a similar to that of solving linear algebraic equations.

One must always do the same to both sides.

Imagine the real number line (x-axis if you are so inclined). One can explain why one is able to add the same value to either side of an inequality through this method. Consider that if one takes the inequality:

a > b ,

and one then adds "c" to either side:

a + c > b + c .

This is evidently still true as one is moving both of the values (represented by either side) along the line of real numbers by "c", and hence the distance between them is the same, and hence the inequality is still true. One must realise that as subtraction can be shown as the addition of a negative number, it is shown (in this explanation) that one can also subtract the same value from either side while allowing the remaining inequality to be true).

Multiplication (and Division, as the inverse function) is different.

It is true that one can multiply by the same positive number on either side of an inequality, however, multiplying by negative numbers has an effect on the inequality symbol.

Consider the following:

c > 0

$a, b, c \in \mathbb{R}$

a > b

What happens if one is to multiply both sides by "-c".

As "a" is greater than "b", then:

ca > cb .

However, it follows that:

-ca < -cb .

This is important. If one is to divide, or multiply both sides of an inequality by a negative value, it will reverse the inequality.

Quadratic inequalities

If one is define a function as a quadratic expression, and then one states that:

f(x) > 0 ,

f(x) < 0 ,

f(x) <= 0 ,

f(x) >= 0

One has created a quadratic inequality.

One does have to be careful when one solves these inequalities, as they generally have a range of possible answers which has two boundaries.

Consider the meaning of the following inequality:

$ax^{2} + bx + c > 0$

If one is to plot the graph of $y = ax^{2} + bx + c$ , one will get a parabolic curve, which may, or may not have any intercepts with the x-axis.

This whole concept is similar to the idea of the solution to an equation of the form:

$ax^{2} + bx + c = 0$

Being the intercepts of the graph of:

$y= ax^{2} + bx + c$

With the x-axis.

The same is true for inequalities, but one has to be more careful about which side of the axis one is observing.

Suppose:

a > 0

$ax^{2} + bx + c > 0$

Values for which this inequality is satisfied are those values of "x" which produce a positive answer (due to the greater than zero symbol). These values are all of the values on the curve:

$y = ax^{2} + bx + c$

Who lie above (and not on) the x-axis.

Now, consider that if the expression has a positive "a", and in other senses is such that it will have two distinct intersections with the x-axis. One can now think of these as the critical values as these are the values upon which the solutions will be based.

In general, the critical values are those values of "x" for which the expression will be equivalent to zero (graphically, these are the x-axis intercepts).

Example

1. Solve the following inequality:

$x^{2} - 6x + 8 > 0$

One must find the critical values; values of "x", such that the following equation is true:

$x^{2} - 6x + 8 = 0$

Hence:

(x - 2)(x - 4) = 0

Therefore the critical values are:

$x = 2, \ or \ x = 4$

Hence one is now aware that graphically, if one was to plot the quadratic expression against "y", one would have x-axis intercepts of "2", and "4".

Now one must consider what the inequality was referring to. The "greater than zero" is important as it shows that the inequality is referring to those values for which the expression is greater than zero (or above the x-axis, graphically).

Now, one can either use a small sketch graph, or a mental sketch graph to solve this inequality, giving:

$x < 2, \ or \ x > 4$ .

It is often advised that students use a small sketch graph for this type of problem, and it is very useful as it does allow for a rather useful, and visual representation of the situation.

One must certainly be vigilant of those quadratic inequalities for which there are no solutions. One can see, evidently, that there will be no values of "x" for which a quadratic expression is below zero, if, in fact, it does not have x-axis intercepts when it is plotted against "y".