Solving One-Step Inequalities

An inequality is a mathematical sentence that compares two expressions using inequality symbols instead of an equal sign. Solving a one-step inequality is very similar to solving a one-step equation: your goal is to isolate the variable.

Inequality Symbols

Before we solve, let's review the symbols:

• $<$ : Less than
• $>$ : Greater than
• $\leq$ : Less than or equal to
• $\geq$ : Greater than or equal to

Solving Using Inverse Operations

To solve a one-step inequality, use the inverse (opposite) operation to get the variable by itself.

• If a number is added to the variable, subtract it from both sides.
• If a number is multiplied by the variable, divide both sides by that number.

(Note: For 6th-grade math, you will mostly work with positive numbers. Later on, you'll learn a special rule for multiplying or dividing inequalities by negative numbers!)

Graphing Inequalities on a Number Line

Because inequalities have an infinite number of solutions (a whole range of numbers), we often represent the solution set on a number line.

1. Choose the right circle:
   • Use an open circle ($\circ$) for $<$ or $>$. This means the starting number is not included in the solution.
   • Use a closed circle ($\bullet$) for $\leq$ or $\geq$. This means the starting number is included.
2. Draw the arrow:
   • Point the arrow to the right for $>$ or $\geq$ (values are getting larger).
   • Point the arrow to the left for $<$ or $\leq$ (values are getting smaller).

Writing Inequalities from Words

Sometimes you need to translate a real-world sentence into math. Look for keywords:

• "More than" $\rightarrow >$
• "Less than" $\rightarrow <$
• "At least" (minimum) $\rightarrow \geq$
• "At most" (maximum) $\rightarrow \leq$

Example Problems

Example 1: Solve and graph $x + 3 > 10$

To isolate $x$, do the inverse of adding 3, which is subtracting 3 from both sides: $x + 3 - 3 > 10 - 3$ $x > 7$

Graphing: Draw an open circle at 7 (because it's just "greater than", not equal to), and draw an arrow pointing to the right to show all numbers larger than 7.

Example 2: Solve and graph $2x \leq 14$

The variable $x$ is being multiplied by 2. The inverse is dividing by 2: $\frac{2x}{2} \leq \frac{14}{2}$ $x \leq 7$

Graphing: Draw a closed circle at 7 (because it can be "equal to" 7), and draw an arrow pointing to the left to show all numbers smaller than 7.

Example 3: Write an inequality for "a number is at least 5"

Let the unknown number be $x$. The phrase "at least" means the number can be exactly 5, or anything greater than 5. Therefore, the inequality is: $x \geq 5$