Solving Equations and Inequalities
An equation consists of two mathematical expressions separated by an equals sign. For instance: 1 + 1 = 2. Any operations can be performed the same way on both sides of the equation and yield the same result. Inequalities consist of two mathematical expressions separated by a sign indicated which side is greater or lesser. For example: 1<2.
Equations and Inequalities
Hi, and welcome to this Mometrix video on equations and inequalities. In this video, we’ll talk about their similarities and differences, what they mean, and we’ll break down, in a way that’s easy to understand, the sometimes confusing inequality signs.
So let’s start with a basic question: What’s an equation?
Here’s a very simple way to think of an equation. An equation has an “equals” sign, like this:
2 + 2 = 4
When you see an equals sign, you know that math problem is an equation. It is saying that two or more things are equal. Those “things” can be simple or complex. Here’s an example of a simple equation:
10 + 2 = 6 + 6
As you can see, the answer on both sides of the equals sign is 12. The equation says that the sum of the numbers on the left side (10+2) equals the sum of the numbers on the right side (6+6).
Equations can be complex, but at their core, either side of the equals sign remains true. Let’s break down this more complex equation, using this example as a base.
5 x 2 ÷ 2 + 10 + 2 = 5 x 2 ÷ 2 + 6 + 6
The answer, on both sides of the equation, is 17. Let’s break it down.
5 x 2 = 10, divided by 2 is 5. 5 plus 10 plus 2 equals 17.
And on the other side:
5 x 2 = 10, divided by 2 = 5.
Then, 5 + 6 + 6 is also 17.
That’s a simple look at equations. Let’s look at some key terms regarding algebraic equations:
An algebraic expression is the problem you’re trying to solve. X + 7 = 14 is a simple algebraic expression. These expressions contain numbers, variables, and an arithmetic operation that can be as simple as addition or subtraction or as complex as a square root multiplier.
A term is at least one number or variable multiplied together, though terms can have more than one number of variable. For example, 2 x – 4 x = 20. The “2” and the “4” are numbers, while “X” is the variable.
Coefficients multiply variables. So, let’s take a look at 6x, which means 6 times the variable “x.” Therefore, the number 6 is the coefficient.
Constants are numbers whose values don’t change. The values are fixed. Let’s look at n + 5 = 9. The values “5” and “9” don’t change, so those are the constants. Constants can also be variables that stand in for fixed numbers.
Like Terms have the same variables and exponents. So, 5xy, 6xy, and 9xy are like terms because they all contain “xy.”
Exponents are the simplified method of multiplication. The “3” in x³ is an exponent. It’s much easier to write and read this m³ + y³ + n² than write and read this:
m x m x m + y x y x y + n x n.
Inequalities also have a large part in algebra. While equations mean two things are equal, inequalities (as you might have guessed) show that things are not equal. These are the 5 inequality signs:
Greater than: >
Greater than or equal to: ≥
Less than: < Less than or equal to: ≤ not equal ≠ So, in simple inequality terms, 8 < 16, 15 > 9, and 6 ≠ 5. Expressed algebraically, you might see:
x > n + 17
Equations and inequalities are similar in some ways, but pretty different in others. Let’s start with the similarities. You can multiply and divide numbers in inequalities in almost the same way you can when working with equations. Here’s an easy example:
9x + 10 > 3x + 4
Equations are true. In other words, the value after the equal sign is absolute. There’s no dispute that 10 + 10 = 20. With inequalities, there are more possible outcomes since there is an infinite number of possibilities for numbers that are less than and greater than.
Negative numbers work differently. This is where it gets a little bit complicated. Anytime you use a negative number to multiply or divide an inequality, you have to “flip” the inequality sign to keep the equation true.
4 > 3.
Four is greater than three. We know that to be true. So let’s expand on both sides a little bit:
3 x 4 > 3 x 3.
In this case, 12 is greater than 9. Once again, that’s true.
But if we multiply with negative numbers, things change a little bit.
Let’s have our equation multiply both sides by – 3: – 3 x 4 > -3 x 3. The answer becomes -12 > – 9.
But that’s not right. We know negative 12 isn’t greater than negative 9.
That’s why you have to reverse the inequality. If you don’t, the problem won’t be true. So reversing the inequality results in -12 < -9, which is true. So that’s our look at equations and inequalities in algebraic equations. Equations present a true value while inequalities can have any number of outcomes. I hope this overview was helpful! Thanks for watching and happy studying!