Solving and Graphing Inequalities

Solving inequalities is similar to solving equations, but with one key difference: inequalities involve a range of possible solutions. Instead of finding a single value for a variable, we find all values that make the inequality true. Multi-step inequalities require combining several steps to isolate the variable.

First, simplify each side of the inequality by distributing and combining like terms. Next, use inverse operations to isolate the variable term on one side. Add or subtract constants from both sides to move constant terms away from the variable. Then, multiply or divide both sides by the coefficient of the variable. Remember, if you multiply or divide by a negative number, you must flip the inequality sign!

For example, solve 3x + 5 < 14. Subtract 5 from both sides to get 3x < 9. Then, divide both sides by 3 to get x < 3. This means any value of x less than 3 will satisfy the original inequality.

To graph the solution on a number line, draw an open circle at 3 (because x is not equal to 3) and shade the line to the left, indicating all values less than 3. If the inequality were x ≤ 3, you would use a closed circle at 3 to show that 3 is included in the solution set. Graphing the solution set provides a visual representation of all possible values that satisfy the inequality.