Question

Explain the differences between properties of equality and properties of inequality when solving equations and inequalities.

Answer 1

An equality is a statement of equal measure. It stands for an absolute statement, without any leeway. That is, there is only a set number of solutions they can take.

Here is an example: x + 17 = 20
In this case, x can only take one solution because it is an absolute statement. Obviously, these can change, but conceptually, they will contain a set of answers a variable can take.

An equality with degree of n will inevitably have n number of answers a variable can take.

However, there are more solutions x can take for an equality. This is because inequality signs are a broader set of equality signs.

Example: x + 17 > 20

In the previous example, there was only one solution that x can take, namely x = 3. However, if we have an inequality, we're merely finding all sets of values x can take that will keep this statement true. In this case, there are an infinite amount of solutions, provided x is greater than 3.

Properties of inequalities vs equalities
This segment is quite tricky to grasp, because we are so used to the equality. The hardest section is to determine when to change the inequality sign. Whenever we multiply or divide by a negative number, we must flip the sign.

Let's consider:
$\frac{x}{x - 3} \geq 9$

We can do this in one of two ways: since we don't know when x is positive or negative, what we know is that if we square a number, it will always be positive.

Hence, $\frac{x(x - 3)^{2}}{x - 3} \geq 9(x - 3)^{2}$ will keep the inequality sign, because we are multiplying by a positive number.

$x^{2} - 3x \geq 9x^{2} - 54x + 81$
$0 \geq 8x^{2} - 51x + 81$

And this becomes a quadratic, you can complete the square or use the quadratic formula.

Answer 2

Using the properties of equality on any equation will not change the solution set. When multiplying or dividing an inequality by a negative number you must reverse the inequality symbol or you will not get the correct solution set.