# 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.

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.

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.
For example:

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!

### What is an equation?

#### A

“An equation shows that two expressions are equal to one another. 4x+3=7”

### What is a linear equation?

#### A

A linear equation is any equation that graphs as a line.

### What is an inequality?

#### A

An inequality is a statement comparing two expressions that do not equal each other. Instead, they use the comparisons less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).
Ex. x > 3

### How do you solve inequalities?

#### A

To solve inequalities, follow the same steps as with an equation. The order of operations is: parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right. To isolate a variable, work through this process backwards, starting with addition and subtraction and ending with parentheses. There is only one slight difference. If you multiply or divide by a negative number, you have to flip the inequality sign.
Ex.
-4x + 3 ≥ 15
-4x ≥ 12
x ≤ -3

## Practice Questions

Question #1:

Simplify the equation:
$$9\times3-11+2=24÷2+18÷3$$

$$18=10$$
$$18=18$$
$$10=10$$
$$40=18$$

Both sides of the equal sign must be the same. Simplify the expressions on either side of the equal sign using the Order of Operations.

We can start by simplifying the expression to the left of the equal sign.
$$9\times3-11+2$$

According to the Order of Operations, we must multiply before adding and subtracting. Since $$9\times3=27$$, rewrite the expression using this product.
$$27-11+2$$

From here, add or subtract from left to right. Since $$27-11=16$$, rewrite the expression using 16.
$$16+2$$

$$16+2=18$$, so the expression to the left of the equal sign is equal to 18.
$$18$$

Now we simplify the expression to the right of the equal sign.
$$18=24\div2+18\div3$$

According to the Order of Operations, we must divide from left to right before adding. Since $$24÷2=12$$, rewrite the expression using this quotient.
$$18=12+18\div3$$

Next, divide $$18÷3$$, which equals 6. Rewrite the expression using this quotient.
$$18=12+6$$

$$12+6=18$$, so the expression to the right of the equal sign is equal to 18. The correct answer is B.
$$18=18$$

Question #2:

What is the name of the highlighted portion in the algebraic equation shown?
$$4x^2-xy-y^2=2$$

Exponent

Constant

Variable

Coefficient

A coefficient is a number that is placed before a variable to multiply it. Since 4 is adjacent to x, it is the coefficient in this equation.

Question #3:

Michelle is working on a school project. She needs to write an inequality to describe the weather forecast for the upcoming week. Wednesday’s temperature will be colder than 30oF. Which inequality best represents this statement?

$$x\geq30$$
$$x\leq30$$
$$x>30$$

$$x$$ < $$30$$

D is the correct answer. In the inequalities given, x represents Wednesday’s temperature. The inequality symbol that can be used to represent “colder than” is the less than (<) symbol. The inequality that best matches the scenario is x < 30.

Question #4:

Natasha’s mom gives her $20 to spend at the fair. The fair admission costs$6 and the rides cost \$0.25 each. To find out how many rides she can go on, Natasha wrote the following inequality:
$$0.25x+6\leq20$$

Based on this inequality, what is the greatest amount of rides Natasha can go on at the fair?

56 rides

55 rides

48 rides

30 rides

To find the answer, solve for x in the inequality.
$$0.25x+6\leq20$$

First, isolate the variable by doing inverse operations, starting with addition and subtraction. Since the inverse of adding 6 is subtracting 6, subtract 6 from both sides of the inequality.
$$0.25x+6-6\leq20-6$$

The sixes cancel out on the left side of the inequality. Since $$20-6=14$$, we can rewrite the inequality using 14.
$$0.25\leq14$$

Next, we isolate the variable by doing inverse operations again. Since the inverse of multiplying is dividing, divide both sides of the inequality by 0.25.
$$0.25\div0.25\leq14\div0.25$$

The left side of the inequality cancels out, leaving us with x. On the right side, we have $$14÷0.25$$ which equals 56. Therefore, x is less than or equal to 56.

The greatest amount of rides Natasha can go on is 56.
$$x\leq56$$

Question #5:

Simplify the following inequality:
$$-4\times8$$ < $$-4\times9$$

$$32$$ < $$-36$$

$$-32$$ < $$-36$$

$$-32>-36$$
$$-36>-32$$
When multiplying or dividing by a negative number, you must reverse the order of the inequality sign. Since both sides of this inequality are multiplied by -4, the inequality sign needs to be flipped from less than (<) to greater than (>). $$-4\times8=-32$$ and $$-4\times9=-36$$. Therefore, the correct answer is C.