Lesson 3: Linear Inequalities

Property

To solve a multi-step linear inequality, follow a systematic flow:

1. Simplify each side completely (distribute and combine like terms).
2. Use the Addition or Subtraction Properties of Inequality to collect all variable terms on one side and all constant terms on the other side.
3. Use the Multiplication or Division Properties of Inequality to isolate the variable. (Remember to reverse the inequality sign if you multiply or divide by a negative number!)

Examples

• Example 1: Solve $3x + 5 > 20$.

Subtract 5 from both sides to get $3x > 15$.
Divide by 3 to get $x > 5$.

• Example 2 (Variables on both sides): Solve $7p - 2 \leq 3p + 10$.

Subtract $3p$ from both sides to gather variables on the left: $4p - 2 \leq 10$.
Add 2 to both sides to gather constants on the right: $4p \leq 12$.
Divide by 4 to get $p \leq 3$.

• Example 3 (Negative division): Solve $5(k - 2) > -20$.

Distribute to get $5k - 10 > -20$.
Add 10 to both sides: $5k > -10$.
Divide by 5 to get $k > -2$. (The sign stays the same because we divided by a positive 5).

Explanation

Solving a multi-step inequality uses the exact same strategy as solving a multi-step equation: clean up both sides, move the letters to one team and the numbers to the other, and then isolate the variable. The only difference is the golden rule of inequalities—you must stay highly alert during the very last step. If you divide or multiply by a negative number to get the variable by itself, you must flip the inequality symbol.

Property

Any number greater than 3 is a solution to the inequality $x > 3$.
We show the solutions to the inequality $x > 3$ on the number line by shading in all the numbers to the right of 3.
The open parentheses symbol, (, shows that the endpoint of the inequality is not included.
The open bracket symbol, [, shows that the endpoint is included.
Because the number 3 itself is not a solution, we put an open parenthesis at 3.
The graph of the inequality $x \geq 3$ is very much like the graph of $x > 3$, but now we need to show that 3 is a solution, too.
We do that by putting a bracket at $x = 3$.
We can also represent inequalities using interval notation.
The inequality $x > 3$ is expressed as $(3, \infty)$.
The symbol $\infty$ is read as 'infinity'.
The inequality $x \leq 1$ is written in interval notation as $(-\infty, 1]$.
The symbol $-\infty$ is read as 'negative infinity'.

Examples

• The inequality $x < 4$ includes all numbers to the left of 4. On a number line, we place a parenthesis at 4 and shade to the left. In interval notation, this is $(-\infty, 4)$.

• The inequality $y \geq -2$ includes -2 and all numbers to its right. On a number line, we place a bracket at -2 and shade to the right. In interval notation, this is $[-2, \infty)$.