Modules for Prerequisite Mathematics

Subsection 3.1.1 Equality and Inequality

I made a big deal in of the idea of equality as a central concept in mathematics, particularly the notion of preserving equality by treating both sides of an equal the same under any operation. We have something very similar for inequalities as well. As a basic reminder, we have four inequality operations.

• Greater than: \(\gt\text{.}\)
• Less than: \(\lt\text{.}\)
• Greater than or equal to: \(\geq\text{.}\)
• Less than or equal to: \(\leq\text{.}\)

For the first two (\(\lt\) and \(\gt\)), if we want to point out that equality is not included, we will often say stricky greater than or less than.

An inequality is any expression involving one of the four inequality opartions. Like an equation, it has a left side and a right side, related by the inequality. For example

describes the set of all numbers (represented by the variable \(y\) such that when we square the number, mutilply by 4 and subtract 9, the result is (strictly) greater than 12.

Working with inequalities is very much like working with equations. In equations, we said we could perform any operation as long as we performed the same operation to both sides of the equation, so that we preserved the equality. Here we have something similar but slightly more restrictive.

We can perform any operation to an inequality as long as it is it performed to both sides of the equation and it is an operation which preserved inequalities.

Subsection 3.1.2 Preserving Inequalities

Here is a list of operations that preserve inequalities, each with a brief example

• Addition and subtraction.
  \begin{align*} 5 \amp \lt 7\\ 5 + 6 \amp \lt 7 + 6 \\ 11 \amp \lt 13 \end{align*}
• Multiplication or division by positive numbers.
  \begin{align*} 5 \amp \lt 7\\ 5 \times 2 \amp \lt 7 \times 2\\ 10 \amp \lt 14 \end{align*}
• Positive whole exponents, as long as both sides are positive.
  \begin{align*} 5 \amp \lt 7\\ 5^2 \amp \lt 7^2\\ 25 \amp \lt 49 \end{align*}
  Note that exponents can easily fail to preserve inequality is one of the sides is not positive.
  \begin{align*} -5 \amp \lt 3\\ (-5)^2 \amp \lt 3^2 \\ 25 \amp \lt 9 \end{align*}
  This is no longer true, since 25 is clearly greater than 9.
• Logarithms, if both terms are positive. (The logarithm isn't even defined if one of the terms is negative)
  \begin{align*} 8 \amp \lt 16\\ \log_2 8 \amp \lt \log_2 16\\ 3 \amp \lt 4 \end{align*}

In the language of Chapter 11 and Chapter 12, any function which is increasing will preserve an inequality when applied to both sides.

Subsection 3.1.3 Reversing Inequalities

Some operations reverse inequalities instead of preserving them. We can also work with these operations when we manipulate inequalities if we remember to reverse the inequalities. For common problems, there are two operations that reverse inequalities.

• Mulitplication or division by negative numbers.
  \begin{align*} 3 \amp \lt 5\\ (3)(-4) \amp \gt (5)(-4)\\ -12 \amp \gt -20 \end{align*}
• Negative exponents, as long as both sides of the inequality are positive.
  \begin{align*} 3 \amp \lt 5\\ 3^{-2} \amp \gt 5^{-2}\\ \frac{1}{3^2} \amp \gt \frac{1}{5^2} \\ \frac{1}{9} \amp \gt \frac{1}{25} \end{align*}
• As a special case of the previous operation, appyling the exponent \(-1\) is the same as taking the reciprocal of both sides. We can think of that as its own operation. Taking the reciprocal of an inequality reverses the inequality. In general, this still requires that the terms are both positive; however, it also works when the terms are both negative. It can fail when one term is positive and one is negative.
  \begin{align*} 3 \amp \lt 5\\ \frac{1}{3} \amp \gt \frac{1}{5} \end{align*}

In the language of Chapter 11 and Chapter 12, any function which is decreasing will reverse an inequality when applied to both sides. However, many operations neither preserve or reverse inequalitys. For example, we cannot usually apply a trigonometric function to an inequailty, since trigonometric functions oscillate up an down. We can't expect any consistent behaviour with these functions applied to inequalities.

Subsection 3.1.4 Square Roots

I want to make a special discussion of square root, since they show up frequently enough that they are worth considering and they present a bit of challenge. Consider this inequality.

We take take the square root of both sides of the inequality.

This looks good: if \(z\) is less than 3, then when we square it, \(z^2\) should be less than 9. However, \(z \lt 3\) also includes all the negative numbers; for many large negative numbers, their square beocmes positive and will be larger than 9. This is a problem. We need another restriction. The result of the square root is, in fact, two inequalities.

Only when \(z\) is between \(-3\) and \(3\) will its square be less than \(9\text{.}\) I can state this generally. If \(x\) is a variable and \(a\) is a positive number, the inequality

is equivalent to the double inequality

For greater than inequalities we have something similar as well. Consider an example first.

If we square root, we get \(t \geq 4\text{.}\) That certainly works; numbers larger than \(4\) have square larger than \(16\text{.}\) However, there are also some negative numbers than have square (which are positive!) which are larger than \(16\text{.}\) We also need \(t \leq -4\text{.}\)

Now I'll state this generally. If \(x\) is a variable and \(a\) is a positive number, then the inequality

is equivalent to the double inequality