Linear Inequalities and Number Lines

For an inclusive inequality (also called a non-strict inequality), the endpoint value is included in the set of possible answers. For a strict inequality, the endpoint is strictly excluded.

When carrying out a numerical comparison, you must calculate both sides independently and then picture their positions on a number line. The number furthest to the right is the greater value.

Examples of valid numerical comparisons:

• $5 < 12$ (5 is less than 12)
• $-8 > -20$ (-8 is greater than -20 because it is further right on a number line)
• $3 \le 3$ (3 is less than or equal to 3; this is true because the values are equal)
• $10 \ge 7$ (10 is greater than or equal to 7; this is true because 10 is greater)

Compare $3 - 15 + 9$ and $11 - 2 \times 8$.

Step 1: Calculate the left-hand side (LHS). $3 - 15 + 9 = -12 + 9 = -3$

Step 2: Calculate the right-hand side (RHS) using correct order of operations. $11 - (2 \times 8) = 11 - 16 = -5$

Step 3: Compare the final values. Because $-3$ is further to the right on a number line than $-5$, $-3$ is greater than $-5$. $-3 > -5$

Solving Linear Inequalities

Why settle for a single exact answer when an equation could have an infinite number of correct solutions? Solving a linear inequality requires the exact same inverse operations as solving a standard algebraic equation. Your goal is still to isolate the variable by balancing both sides of the statement.

However, there is one crucial difference known as the "Negative Rule". If you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign to keep the mathematical statement true. Adding or subtracting numbers does not change the sign's direction.

Solve $10 - 2x \le 4$.

Step 1: Subtract $10$ from both sides to isolate the $-2x$ term. $-2x \le -6$

Step 2: Divide both sides by $-2$. Because we are dividing by a negative number, the sign must flip from $\le$ to $\ge$.

$x \ge 3$

Representing Inequalities on a Number Line

How do you clearly show an infinite range of possible numbers on a single piece of paper? A number line is the best visual tool for showing an inequality's solution set. We use distinct circles to mark the starting point and arrows to show the direction of the valid values.

An open circle represents a strict inequality ($<$ or $>$), showing the starting number itself is excluded. A closed circle (which is shaded completely solid) is used for an inclusive inequality ($\le$ or $\ge$) to show the exact endpoint is allowed.

Finally, a horizontal arrow pointing to the right covers "greater than" values, while an arrow pointing to the left covers "less than" values.

Represent the solution $x \ge 3$ on a number line.

Step 1: Identify the endpoint and circle type. At $3$, the symbol is $\ge$ (inclusive), so draw a closed circle.

Step 2: Draw the arrow. Because the solution is "greater than or equal to", draw a horizontal arrow extending to the right from the closed circle.