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Inequalities are all around us—whether it’s checking if you're tall enough for a roller coaster or comparing how much allowance you’ve saved up.
Instead of just saying two things are equal, inequalities let us know exactly when one number is bigger, smaller, or even within a certain range. They help us compare numbers and understand limits.
Read on for simple definitions, easy-to-follow guides, worked-out examples, and a fun quiz to test what you’ve learned!
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In math, we often use the equal sign (=) to show that two numbers or expressions are exactly the same, like 5 = 5 or 2 + 3 = 5.
But what if one number is bigger or smaller than another?
That’s when we use the sign of inequality (≠) instead of the equal sign.
Inequalities in math are statements that compare two values or expressions using special symbols to show when one is greater than, less than, or not equal to the other.
Inequalities help us understand limits, compare values, set limits, and solve problems. They allow us to work with ranges of numbers instead of just single values.
They also show up in everyday life more often than you think!
Minimum age requirements for movies, ingredient limits in recipes, time restrictions for video game sessions, and scoring requirements to pass a test all involve inequalities in some way.
Mastering them is the key to building solid math foundations, and with the right approach, they’re easy to learn and apply.
We’ve already mentioned the inequality signs, so you must be wondering: “How are inequality symbols different from inequality signs?”
It’s easy to confuse the two as they are not that different and are often used interchangeably.
An inequality symbol refers to the mathematical notation we use to compare two values or expressions. So it is how we write the comparison.
The term inequality sign describes the direction or meaning of the inequality in a given problem. For example, when solving an inequality, you may need to flip the sign if you multiply or divide by a negative number. In this sense, the symbol represents the comparison, while the sign indicates the relationship between the values and can change during problem-solving.
The inequality symbols are:
\( \displaystyle > \) (read: Greater than) – The value on the left is larger than the value on the right. For example, if \( \displaystyle x > 3 \), then \( \displaystyle x \) can be any value above 3, including 3.1, 4, 5, and so on.
\( \displaystyle < \) (read: Less than) – The value on the left is smaller than the value on the right. For example, if \( \displaystyle y < 10 \), then \( \displaystyle y \) can be any value below 10, including 9.9, 8, 7, etc.
\( \displaystyle \geq \) (read: Greater than or equal to) – The value on the left is either larger than or exactly equal to the value on the right. For example, if \( \displaystyle a \geq 5 \), then a can be any value including or higher than 5, which can be 5, 6, 7, etc.).
\( \displaystyle \leq \) (read: Less than or equal to) – The value on the left is either smaller than or exactly equal to the value on the right. For example, if \( \displaystyle b \leq 2 \), then \(\displaystyle b \) can include 2 and numbers smaller than 2 like 1, -1, -2, and so on.
\( \displaystyle \neq \) (read: Not equal to) – The two values are different; they are not the same. If \( \displaystyle c \neq 4 \), then \( \displaystyle c \) can be any number except 4.
Comparing numbers is something we do all the time—who got the highest test score, who ran the fastest mile, or who has the biggest pile of candy after trick-or-treating.
Inequalities help us measure, compare, and set limits in math, just like we do in real life.
But not all inequalities work the same way. Some comparisons strictly exclude a number, while others include it.
Let’s explore the different types of inequalities and how they shape the way we solve math problems.
Understanding different types of inequalities is important because they affect what values are possible in a solution.
Strict inequalities use the less than (\( \displaystyle < \) ) and greater than (\( \displaystyle > \)) symbols. These comparisons do not include the number itself.
For example, if \( \displaystyle x < 5 \), that means x can be 4, 3, 2, or anything smaller than 5—but never 5 itself.
Imagine you’re entering a video game tournament, but only players who have scored more than 1,000 points can compete. If you scored exactly 1,000, you don’t qualify! Your score must be strictly greater than 1,000 (score \( \displaystyle > \) 1,000) to enter.
Or, let’s make it simple and just use inequality to describe the world around us, like comparing two bowls of fruit. If one has 4 and another 2 pieces of fruit, one has more (\( \displaystyle > \)) than the other, right?
Non-strict inequalities use less than or equal to (\( \displaystyle \leq \)) and greater than or equal to (\( \displaystyle \geq \)) signs. This means the number can be included in the solution.
For example, if \( \displaystyle x \leq 5 \), then x can be 5 or any number smaller than 5.
A real-life example would be baking cookies. If the recipe says to bake cookies for at least 10 minutes (baking time \( \displaystyle \geq \) 10, read: greater than or equal to 10), it means that we need to bake them for 10 minutes exactly or longer. If we take them out at 9 minutes, they might still be doughy!
A linear inequality is like a regular equation, but instead of an equal sign (\( \displaystyle \neq \)), it uses inequality symbols:
\( \displaystyle > \) (greater than)
\( \displaystyle < \) (less than)
\( \displaystyle \geq \) (greater than or equal to)
\( \displaystyle \leq \) (less than or equal to)
Think of it as a way to compare numbers or expressions! Instead of saying something is exactly equal, we say it's bigger or smaller than another value.
Linear inequalities are solved just like equations—we perform the same steps as we would with a regular equation but replace the equal sign (\( \displaystyle \neq \)) with an inequality symbol.
Let’s look at an example where \( \displaystyle 2x + 3 \) is greater than or equal to \( \displaystyle 7 \).
\( \displaystyle 2x + 3 \geq 7 \)
\( \displaystyle 2x \geq 7 - 3 \)
\( \displaystyle 2x \geq 4 \)
\( \displaystyle x \geq 4 \div 2 \)
\( \displaystyle x \geq 2 \)
The result of our expressions with linear inequality is that \( \displaystyle x \) is greater than or equal to \( \displaystyle 2 \).
Compound inequalities combine two conditions into one statement using the words "and" or "or."
Let’s see how they work!
An "and" inequality means a number must satisfy both conditions at the same time.
Think of it like trying to get into a special math club with strict age rules. If the rule says you must be between 3 and 8 years old, that means you can be 3, 4, 5, 6, or 7—but not 8 or older. You have to be old enough to be inside the allowed range.
Mathematically, we would write this age rule as:
\( \displaystyle 3 \leq x < 8 \)
The symbol \( \displaystyle \leq \) shows that \( \displaystyle x \) can be 3, but the symbol \( \displaystyle < \) means \( \displaystyle x \) cannot be 8. Any number in the range of 3 to 7.9 is a solution.
An "or" inequality is different.
A compound inequality with "or" means that at least one of the inequalities must be true. A solution is valid if it satisfies either one of the inequalities or both.
It’s like choosing a ride at an amusement park. Some rides are only for kids under 5, while others like roller coasters are for teens over 12. If you’re 4 or younger, you ride the under-5 rides. If you’re 13 or older, you get to ride the big roller coasters. But if you’re between 5 and 12, you don’t qualify for either!
Mathematically, we would write this as:
\( \displaystyle x < 5 \quad \text{or} \quad x > 12 \)
This means x can be either less than 5 or greater than 12.
Unlike "and" inequalities, where numbers must fit within a specific range, "or" inequalities allow two separate sets of solutions. A number like 3 satisfies \( \displaystyle x < 5 \), and a number like 15 satisfies \( \displaystyle x > 12 \).
Now we are going into high-school-level material, so if you do not understand some of the terms, don’t worry. You can skip ahead to the rules for inequalities.
A quadratic inequality is an inequality that involves a quadratic expression, meaning the variable is raised to the second power (\( \displaystyle x^2 \)). Instead of a single solution, quadratic inequalities often have a range of values as solutions.
An example of a quadratic inequality would be:
\( \displaystyle x^2 - 4x - 5 > 0 \)
Unlike the ‘regular’ quadratic expressions where we are looking for one solution, in a quadratic one we’re looking for a range of values that make the inequality true.
Just like regular quadratic equations, quadratic inequalities can be solved by factoring, using the quadratic formula, or graphing. The key difference is that, instead of finding just one answer, we look at where the graph is above or below a certain point.
It’s like playing basketball with your friends and you throw the ball into the air. Its height over time forms an arch that follows a quadratic equation.
A quadratic inequality can help answer a question like, “When is the ball higher than 10 feet?” Instead of one moment, you’d find a whole range of times when the ball is above that height.
Inequalities have their own special set of rules, just like a board game or a sports game. Once we know all the rules, we can solve math problems correctly and more quickly.
When you add or subtract the same number from both sides of an inequality, the inequality sign stays the same—no flipping needed!
For example:
\( \displaystyle x + 3 < 7 \)
To isolate and solve for \( \displaystyle x \), we can subtract 3 from both sides:
\( \displaystyle x + 3 - 3 < 7 - 3 \)
\( \displaystyle x < 4 \)
When we subtract 3 from both sides, the answer is 4 and the sign remains unchanged, so the \( \displaystyle x \) is less than 4, which can be 3, 2, 1, or even -100.
When we are working with inequalities, multiplying or dividing both sides by a number follows the same rules as equations, but with one important difference: if you multiply or divide by a negative number, we must flip the inequality sign!
Let’s start simple, with a multiplication by a positive number which means that our sign remains the same throughout:
\( \displaystyle 2x < 8 \)
To isolate and solve for x, we can divide both sides by 2.
\( \displaystyle \frac{2x}{2} < \frac{8}{2} \)
\( \displaystyle x < 4 \)
Since we were dividing by a positive number (2), our sign remained unchanged and the result tells us that \( \displaystyle x \) is less than 4.
Now let’s look at an example of dividing by a negative number.
\( \displaystyle -2x < 8 \)
To isolate the \( \displaystyle x \), we divide both sides by -2.
\( \displaystyle \frac{-2x}{-2} < \frac{8}{-2} \)
\( \displaystyle x > -4 \)
Notice how we flipped the sign? Because we divided by a negative number (-2), we flipped the sign in the result.
The transitive property tells us that if one number is smaller than a second number, and that second number is smaller than a third number, then the first number must also be smaller than the third.
Let’s see this in math form:
If \( \displaystyle \mathbf{x < y} \) and \( \displaystyle \mathbf{y < z} \), then it must be true that \( \displaystyle \mathbf{x < z} \).
Let’s look at an example:
Imagine three students comparing their heights:
Max is shorter than Leo (Max \( \displaystyle < \) Leo)
Leo is shorter than Oliver (Leo \( \displaystyle < \) Oliver)
Since Max is shorter than Leo and Leo is shorter than Oliver, that must mean Max is also shorter than Oliver! (Max \( \displaystyle < \) Oliver)
This rule is helpful when solving multiple inequalities at once. It helps us connect comparisons and find the biggest or smallest values in a group.
Now it’s time to put our inequality skills to the test.
Let’s work through some problems step by step and see how inequalities help us compare numbers, find missing values, and make sense of mathematical relationships.
Let’s solve the inequality step by step and understand what each step means.
Solve for \( \displaystyle x \):
\( \displaystyle x + 5 \leq 12 \)
The inequality \( \displaystyle x + 5 \leq 12 \) means that when you add \( \displaystyle 5 \) to \( \displaystyle x \), the result must be less than or equal to \( \displaystyle 12 \). Our goal is to isolate \( \displaystyle x \) so we can see what values it can take.
To isolate the x, we need to remove the +\( \displaystyle 5 \) on the left side, so we simply subtract \( \displaystyle 5 \) from both sides of the inequality.
\( \displaystyle x + 5 - 5 \leq 12 - 5 \)
This simplifies to:
\( \displaystyle x \leq 7 \)
The solution \( \displaystyle x \leq 7 \) means that \( \displaystyle x \) can be\( \displaystyle 7 \) or any number smaller than \( \displaystyle 7 \). So, possible values for x include 7, 6, 5, 4, 3, 2, 1, 0, and even negative numbers.
Let’s solve the inequality:
\( \displaystyle 3x > 18 \)
The inequality \( \displaystyle 3x > 18 \) means that three times some number \( \displaystyle x \) is greater than \( \displaystyle 18 \). Our goal is to figure out what values of \( \displaystyle x \) make this statement true.
To isolate and solve for \( \displaystyle x \), we will divide both sides by \( \displaystyle 3 \):
\( \displaystyle 3x \div 3 > 18 \div 3 \)
This simplifies to:
\( \displaystyle x > 6 \)
The solution \( \displaystyle x > 6 \) means that \( \displaystyle x \) can be any number greater than 6, like 7, 10, or even 1000. However, \( \displaystyle x \) cannot be 6 or anything smaller, since the inequality does not include "or equal to" \( \displaystyle \geq \).
Let’s look at that case where dividing (or multiplying) by a negative number means that we have to flip the inequality sign in the answer.
\( \displaystyle -4x < 16 \)
To isolate the x, we divide both sides of the inequality by -4:
\( \displaystyle \frac{-4x}{-4} < \frac{16}{-4} \)
Whenever we divide or multiply by a negative number, we must flip the inequality sign.
Since we divided by -4, the "<" sign flips to ">":
\( \displaystyle x > -4 \)
This means that \( \displaystyle x \) can be any number greater than -4.
For example, numbers like -3, 0, or 10 would all be solutions, but numbers like -5 or -6 would not work because they are smaller than -4.
Always remember—if you multiply or divide by a negative number, flip the inequality sign. This is one of the most common mistakes students make, so keep an eye out for it!
Now that we've mastered the basics of inequalities, it’s time to put your skills to the test!
Solve the below:
Solve for x: \( \displaystyle x + 7 \leq 12 \)
Solve for x: \( \displaystyle x - 5 > 3 \)
Solve for x: \( \displaystyle 4x < 20 \)
Solve for x: \( \displaystyle -3x \geq 9 \)
Solve for x: \( \displaystyle x - 2 < 6 \)
Which of the following is equivalent to the inequality: \( \displaystyle -8 \geq 2x \)
\( \displaystyle x \geq 10 \)
\( \displaystyle x \leq 10 \)
\( \displaystyle x \geq -6 \)
\( \displaystyle x \leq -6 \)
If \( \displaystyle x > -4 \), which of the following is true after multiplying both sides by -3?
\( \displaystyle x > 12 \)
\( \displaystyle x < 12 \)
\( \displaystyle x > -12 \)
\( \displaystyle x < -12 \)
Solve for x: \( \displaystyle -5x + 10 > 0 \)
When you finish solving the exercises, check your answers at the bottom of the guide.
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Our specially trained math tutors provide personalized instruction, including in-person and online support, to help students truly understand and enjoy math.
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If you’ve given our exercises a try, check your answers here:
\( \displaystyle x \leq 5 \) (Subtract 7 from both sides)
\( \displaystyle x > 8 \) (Add 5 to both sides)
\( \displaystyle x < 5 \) (Divide both sides by 4)
\( \displaystyle x \leq -3 \) (Divide by -3 and flip the sign)
\( \displaystyle x > -12 \) (Multiply by -2 and flip the sign)
A) \( \displaystyle x \geq 10 \) (Add 8 to both sides)
B) \( \displaystyle x < 12 \) (Multiply by -3 and flip the sign)
\( \displaystyle x < 2 \) (Subtract 10, then divide by -5 and flip the sign)
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