Roger Layton

Today I will focus on the questions in the Grade 12 examination papers which include inequalities.

Let me first distinguish between an equation and an inequality.

An equation is a mathematical statement, mostly with two sides (left hand side and right hand side) which declares that both sides are equal. Equations are fundamental to all of mathematics and are statements that the equality between the two sides holds as a universal truth, perhaps with some conditions in which is may be true.

Let us consider two examples of equations, firstly the Theorem of Pythagoras:

$a^2 = b^2 + c^2$

which states the relationship between the sizes of the sides of a right-angled triangle, where a is the hypotenuse (the longest side), and b and c are the other two sides. This holds for every right-angled triangle, but not for other triangles.

Another famous example, perhaps the most quoted equation of all time, is Einstein’s formula

$E = mc^2$

However, there are also a range of mathematical statements which do not indicate quality between the sides, but rather than inequality. These are statements in which the one side is less-than the other side, or is less-then-or-equal-to the other side. These are expressed using the following symbols

$<$

$\leq$

$>$

$\geq$

So consider a single example of an inequality

$x\geq{4}$

which states that x has a value which is greater than or equal to 4. However, this is a statement, and this statement can either be true or false, depending on the value of $x$ . So if $x=2$ then this statement is false, and if $x=5$ then it is true. So what if $x=4$ ? This is where the difference lies between $>$ and $\geq$ . In the pure greater-than case, then 4 will NOT make this statement true. However, in the greater-than-or-equal case, then the value of 4 will make this statement true.

This statement is too simple for our purposes here, and we need to consider more complex examples, similar to what are provided in the Grade 12 examinations. For example consider the following question:

Solve for $x$ where

$(x-2)(x+4)>0$

which could also be stated, equivalently, as

$x^2+2x-8>0$

by just multiplying out the factors. So both of these forms are the same, yet the second requires the additional step of factorisation to support an approach to its solution.

You may have been taught procedures to solve this type of problem, but I would rather resort to first principles and solve this by inspection.

The first principle that is for

$ab>0$

Then either both are positive

$a>0$ and $b>0$

or both are negative

$a<0$ and $b<0$

There is no other way to create a positive result when the two values are multiplied, since if one is negative and the other is positive this will always yield a negative result. If one or both of the values are zero, their product is zero., and thus neither positive or negative.

So getting back to our problem, let us look at the first case, in which both case positive, which means that

$x-2>0$ and $x+4>0$

In the first expression, $x$ must be greater than 2 for this to be positive, and for the second this requires $x>-4$ .

Thus if $x>2$ then both of these are satisfied, both values are positive, and the expression is positive.

In the second case, both must be negative which means that

$x-2<0$ and $x+4<0$

Thus $x<2$ or $x<-4$ , so if $x<-4$ then both are satisfied, and this is sufficient for the answer.

This type of problem is typically only given 2 marks, and so you have to work fast to get these marks and to write down the minimum needed to show you have selected an approach which works and also that you have arrived at the correct answer.

Be careful to note whether this is indicated as greater-than, or greater-than-or-equal-to since this will also have an impact on your answer and a silly mistake can cost you the marks.

Have a look at this question from the Grade 12 supplementary exam in Feb-Mar 2017.

Welcome back!

I have included a detailed solution of two of the questions from the Grade 12 supplementary examination from Feb-Mar 2017.

These are part of the Question 1 on Paper 1 which has been consistent for many years.

These questions draw on your prior knowledge of algebra from previous grades, and include you proficiency in factoring numbers, exponents, signs, and basic algebraic manipulation. However, these are all solvable, and you should be able to gain time in the examination by attending to these questions at the start of the three hours that you have available.

Try these two to get started from this examination:

1.1 Solve for $x$ :

1.1.1 $(x-3)(x-1)=0$