Solving Inequalities

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions that are not necessarily equal, using symbols to show their relative size. While an equation uses an equal sign $=$ to state that two things are exactly the same, an inequality tells us how two quantities relate to each other when they might be different. For example, if you need to be at least $16$ years old to drive, we can write that as $age\ge 16$. This single statement covers ages $16,17,25,$ and even $100$.

Inequalities are powerful because they don't just describe a single value; they describe a whole range of possible values. This makes them essential for representing real-world constraints, like speed limits, budget caps, or temperature ranges.

There are four primary inequality symbols you must know:

The key difference to remember is that an equation like $x+3=5$ has only one solution ($x=2$), but an inequality like $x+3>5$ has an infinite number of solutions (any number greater than $2$). Our goal is to find and describe that entire set of solutions.

How Do You Graph Inequalities on a Number Line?

Since inequalities have an infinite range of solutions, we can't just list them all. Instead, we visualize them by graphing on a number line. This gives us a clear picture of every possible number that makes the inequality true.

The process involves two key decisions: the type of circle to use and which direction to shade.

1. Open vs. Closed Circles: The circle at the starting number (the endpoint) tells you whether that number itself is included in the solution.
   • Use an open circle (an unfilled circle: ○) for $<$ (less than) and $>$ (greater than). This signifies that the endpoint is not part of the solution. For $x>3$, the number $3$ itself is not a solution.
   • Use a closed circle (a filled circle: ●) for $\le$ (less than or equal to) and $\ge$ (greater than or equal to). This signifies that the endpoint is part of the solution. For $x\ge 3$, the number $3$ is a valid solution.
2. Shading Direction: The shaded arrow shows all the other numbers that are part of the solution set.
   • If the variable is less than a number ($x<5$ or $x\le 5$), you shade to the left on the number line.
   • If the variable is greater than a number ($x>5$ or $x\ge 5$), you shade to the right on the number line.

For example, to graph $x\le -1$, you would place a closed circle on $-1$ (because of the "or equal to" part) and shade a line extending to the left, covering all numbers less than $-1$.

What Is the Most Important Rule for Solving Inequalities?

Solving an inequality is very similar to solving an equation. You use inverse operations to isolate the variable. You can add the same number to both sides, subtract the same number from both sides, and multiply or divide both sides by the same positive number. However, there is one critical rule that is unique to inequalities.

This is the single most common place where students make mistakes, so it's worth understanding why it works. Let's start with a true statement: $4<10$. This is obviously correct.

Now, let's multiply both sides by $-2$. On the left, we get $4\times -2=-8$. On the right, we get $10\times -2=-20$.

How do $-8$ and $-20$ relate? On a number line, $-8$ is to the right of $-20$, which means $-8$ is greater than $-20$. So, we must write $-8>-20$.

Notice what happened. We started with $<$ and ended with $>$. Multiplying by a negative number reversed the relationship between the two sides. The same logic applies to division. Forgetting this rule will lead to a completely incorrect set of solutions.

How Do You Solve Basic Inequalities?

Solving one-step and two-step inequalities follows the exact same logic as solving equations: use inverse operations to isolate the variable. The goal is to get the variable by itself on one side of the inequality sign. Just remember the Golden Rule about multiplying or dividing by negatives!

The general steps are:

1. Identify the operations being performed on the variable.
2. Apply the inverse operations to both sides of the inequality to undo them, following the reverse order of operations (SADMEP: Subtraction/Addition, then Division/Multiplication).
3. Remember to flip the inequality sign if you multiply or divide by a negative number.

What About Inequalities with Variables on Both Sides?

When inequalities get more complex, involving distribution, like terms, and variables on both sides, the strategy remains the same: simplify and isolate. The goal is still to get all the variable terms on one side and all the constant terms on the other.

Here is a reliable plan of attack:

1. Simplify each side: Use the distributive property to remove parentheses and combine any like terms on the left side and on the right side.
2. Move the variables: Add or subtract terms to get all the variable terms on one side of the inequality. It's often easier to move the variable with the smaller coefficient to avoid negatives, but either way works.
3. Move the constants: Add or subtract terms to get all the constant terms to the opposite side of the inequality.
4. Solve: Perform the final multiplication or division to isolate the variable. Remember to flip the sign if you multiply or divide by a negative number.

What Are Compound Inequalities?

A compound inequality is made up of two inequalities joined by the word "and" or the word "or".

"And" Inequalities (Intersection)

An "and" inequality is true only if both conditions are met simultaneously. These are often written as a single statement, like $-2\le x<5$. This means "$x$ is greater than or equal to $-2$" AND "$x$ is less than $5$". The solution is the overlap, or intersection, of the two individual solution sets.

To solve an "and" inequality like $-3<2x+1\le 7$, you perform the same operation on all three parts to isolate $x$ in the middle.

$$-3-1<2x+1-1\le 7-1$$ $$-4<2x\le 6$$ $$\frac{-4}{2}<\frac{2x}{2}\le \frac{6}{2}$$ $$-2<x\le 3$$

The graph of an "and" inequality is a line segment between the two endpoints. In this case, it would be an open circle at $-2$, a closed circle at $3$, and the line shaded between them.

"Or" Inequalities (Union)

An "or" inequality is true if at least one of the conditions is met. The solution is the combination, or union, of all values that satisfy either inequality. These are always written as two separate statements, like $x<0$ or $x\ge 4$.

To solve an "or" inequality, you solve each part separately.

For example, solve $3n+1<-8$ or $-2n\le -10$.

First part:
$$3n<-9$$
$$n<-3$$

Second part:
$$-2n\le -10$$ Divide by $-2$ and flip the sign! $$n\ge 5$$

The final solution is $n<-3$ or $n\ge 5$. The graph consists of two separate rays: one starting with an open circle at $-3$ and shading left, and another starting with a closed circle at $5$ and shading right.

What Are Common Mistakes When Solving Inequalities?

Solving inequalities requires careful attention to detail. A small mistake can lead to a completely wrong answer. Here are the most common errors to watch out for:

• Forgetting to flip the sign. This is the number one mistake. You must flip the inequality symbol only when you multiply or divide both sides by a negative number.
• Flipping the sign unnecessarily. Some students get confused and flip the sign when they add or subtract a negative number. Remember, the rule only applies to multiplication and division. Solving $x+5>2$ by subtracting $5$ gives $x>-3$; the sign does not change.
• Mixing up open and closed circles. A closed circle (●) means "or equal to" ($\le ,\ge$) and includes the endpoint. An open circle (○) means the endpoint is not included ($<,>$). Double-check the symbol before you graph.
• Shading in the wrong direction. After you've found your endpoint and circle type, make sure you shade correctly. A good habit is to read your final answer aloud: "$x$ is greater than $4$" clearly means you need to shade the numbers larger than $4$, which are to the right on the number line.
• Misinterpreting compound inequalities. Students sometimes graph an "or" inequality as an intersection or an "and" inequality as two separate rays. Remember: "and" means overlap (a segment), while "or" means both parts combined (two opposite rays).

Quick Reference: Key Concepts for Inequalities

When you're studying or need a quick reminder, use this summary of the most important concepts for solving inequalities.