Linear Inequalities

What Is a Linear Inequality?

A linear inequality is a mathematical statement that compares two linear expressions using an inequality symbol. Unlike a linear equation, which uses an equals sign (=) and typically has one specific solution, a linear inequality describes a range of possible values for the variable.

Think of it this way: an equation is like a specific address, while an inequality is like an entire neighborhood. Instead of finding just one number that makes the statement true, you're finding all the numbers that work.

The core of an inequality is its symbol. There are four main symbols you'll work with:

The expressions on either side of the symbol are linear, meaning the variable has an exponent of 1. For example, $3x-1>8$ is a linear inequality, but $x^2+2\le 6$ is not (it's a quadratic inequality).

How Do You Graph Inequalities on a Number Line?

Since the solution to an inequality is a range of numbers, we can't just list them all. The best way to visualize the solution set is by graphing it on a number line. This gives a clear picture of every possible value the variable can take.

There are two key components to graphing an inequality:

1. The Point: We place a circle on the number line at the boundary value. The type of circle depends on the inequality symbol.
   • An open circle (or hollow dot) is used for $<$ (less than) and $>$ (greater than). This shows that the boundary number itself is not part of the solution.
   • A closed circle (or solid dot) is used for $\le$ (less than or equal to) and $\ge$ (greater than or equal to). This shows that the boundary number is included in the solution.
2. The Shading: We shade the part of the number line that contains all the solution values.
   • For "less than" ($<$ or $\le$), you shade to the left of the point.
   • For "greater than" ($>$ or $\ge$), you shade to the right of the point.

What Are the Rules for Solving Linear Inequalities?

Solving a linear inequality is very similar to solving a linear equation. Your goal is to isolate the variable on one side of the statement. You can use the same inverse operations: addition, subtraction, multiplication, and division. However, there is one crucial rule that makes inequalities different.

Here are the properties of inequalities:

• Addition and Subtraction Property: You can add or subtract the same number from both sides of an inequality, and the inequality remains true. The symbol does not change.
If $a>b$, then $a+c>b+c$ and $a-c>b-c$.
• Multiplication and Division Property (Positive Numbers): You can multiply or divide both sides of an inequality by the same positive number, and the inequality remains true. The symbol does not change.
If $a>b$ and $c>0$, then $ac>bc$ and $\frac{a}{c}>\frac{b}{c}$.
• The Golden Rule: Multiplication and Division Property (Negative Numbers): This is the most important rule to remember. When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality symbol.
If $a>b$ and $c<0$, then $ac<bc$ and $\frac{a}{c}<\frac{b}{c}$.

Why do we flip the sign? Consider the true statement $8>4$. If we divide both sides by $-2$, we get $-4$ on the left and $-2$ on the right. On a number line, $-4$ is to the left of $-2$, so $-4<-2$. To keep the statement true, we had to flip the $>$ symbol to a $<$ symbol.

How Do You Solve One-Step and Two-Step Inequalities?

Let's apply the rules to solve some basic inequalities. The process involves using inverse operations to get the variable by itself, just like with equations. Always remember to check if you need to flip the sign.

Solving Multi-Step Inequalities with Variables on Both Sides

More complex inequalities may involve the distributive property, combining like terms, and variables on both sides of the symbol. The strategy remains the same: simplify both sides first, then use inverse operations to isolate the variable.

Here is a general plan of attack:

1. Use the distributive property to remove any parentheses.
2. Combine like terms on each side of the inequality separately.
3. Move all variable terms to one side of the inequality (using addition or subtraction).
4. Move all constant terms to the other side of the inequality.
5. Solve for the variable (remembering to flip the sign if you multiply or divide by a negative!).

What Are Compound Inequalities?

A compound inequality joins two separate inequalities with the word "and" or "or".

'And' Inequalities (Intersection)

An "and" inequality means a value must satisfy both conditions at the same time. These are often written as a single, compact statement.

For example, $x>-1$ and $x\le 4$ can be written as $-1<x\le 4$. This means $x$ is simultaneously greater than $-1$ and less than or equal to $4$.

When you graph an "and" inequality, you are looking for the intersection, or overlap, of the two individual graphs. The graph is typically a line segment between the two boundary points. For $-1<x\le 4$, you would have an open circle at $-1$, a closed circle at $4$, and the line segment between them would be shaded.

'Or' Inequalities (Union)

An "or" inequality means a value must satisfy at least one of the two conditions. It can satisfy one, the other, or both (though in most cases the solution sets are separate).

For example: $k<-2$ or $k\ge 3$.

A number like $-5$ is a solution because it is less than $-2$. A number like $6$ is a solution because it is greater than or equal to $3$. A number like $0$ is not a solution because it does not satisfy either condition.

When you graph an "or" inequality, you graph each part and keep everything you've shaded. This is the union of the two sets. The graph usually consists of two rays pointing in opposite directions. For $k<-2$ or $k\ge 3$, you would have an open circle at $-2$ with shading to the left, and a closed circle at $3$ with shading to the right.

Common Mistakes to Avoid with Linear Inequalities

Inequalities can be tricky, but most errors come from a few common misunderstandings. Be on the lookout for these pitfalls:

• Forgetting to Flip the Sign: This is the number one mistake. Any time you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. Always double-check this step.
• Flipping the Sign Incorrectly: Remember, the flip rule only applies to multiplication and division by negatives. It does not apply when you add or subtract a negative number from both sides.
• Open vs. Closed Circles: Using the wrong type of circle on the number line changes the meaning of your answer. Remember: if the number is part of the solution ($\le$ or $\ge$), the circle is closed. If it's just a boundary ($<$ or $>$), the circle is open.
• Incorrect Shading Direction: A simple way to check your shading is to pick a test point. For the inequality $x>2$, you might test $x=3$. Is $3>2$? Yes, so your shading should cover the number $3$.
• Reading Compound Inequalities Backwards: When an inequality is written like $5>x$, it's easy to get confused. It's often helpful to rewrite it with the variable on the left: $x<5$. This makes it clearer that you need to shade to the left.

Quick Summary and Key Properties

Here's a quick reference guide to the most important concepts for solving and graphing linear inequalities.

Core Goal

• The primary goal is to isolate the variable on one side of the inequality symbol.

Solving Steps

1. Simplify both sides of the inequality (distribute, combine like terms).
2. Move all variable terms to one side and all constant terms to the other using addition or subtraction.
3. Solve for the variable using multiplication or division.

The Golden Rule

If you multiply or divide both sides by a NEGATIVE number, you MUST FLIP the inequality symbol.

• $>$ becomes $<$
• $<$ becomes $>$
• $\ge$ becomes $\le$
• $\le$ becomes $\ge$

Graphing on a Number Line

• Open Circle ( O ): Use for $<$ and $>$. The endpoint is NOT included.
• Closed Circle ( ● ): Use for $\le$ and $\ge$. The endpoint IS included.
• Shade Left: For "less than" solutions ($x<...$ or $x\le ...$).
• Shade Right: For "greater than" solutions ($x>...$ or $x\ge ...$).