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Inequalities, like equations, involve expressions on two sides of a sign. Instead of using an equal sign, inequalities use symbols like \(<\), \(>\), \(\leq\), and \(\geq\). These symbols represent whether one side of the inequality is less than, greater than, less than or equal to, or greater than or equal to the other side. To solve an inequality like \(3x + 2 \leq 14\), you follow similar steps to solving equations. First, you need to isolate the variable. In this exercise, you isolate \(x\) by subtracting 2 from both sides, giving \(3x \leq 12\).

You then divide each side by 3 to solve for \(x\), resulting in \(x \leq 4\). Remember, whenever you multiply or divide by a negative number in inequalities, you must also flip the inequality symbol. However, that rule didn't apply in this example since only positive numbers were involved.

Once you have solved for \(x\), you're ready for the next step, graphing your solution.

Graphing the solution to an inequality helps you visualize the range of numbers that satisfy it. Once you have determined that \(x \leq 4\), the next task is to represent this visually. This is done by placing the solution set onto a graph or a number line. Graphs are great tools for showing the relationship between numbers and often make it easier to understand complex math concepts like inequalities.

For our inequality \(x \leq 4\), the graph will show the point 4 as well as an area that extends to the left of this point. This is because \(x\) includes all numbers less than or equal to 4. Always remember that filled dots or circles indicate the number itself is included in the solution set. In the next section, we'll illustrate how to show this on a number line.

Number lines are simple yet effective tools to represent solutions to inequalities. Start by drawing a horizontal line and labeling some numbers on it. In this case, since \(x \leq 4\), you will highlight the point at 4 on the line. Place a solid dot at 4 to indicate that the number 4 itself is part of the solution. This solid dot signifies 'included' because of the "less than or equal to" symbol.

Next, draw an arrow extending to the left from the dot at 4. This arrow represents all numbers less than 4, showing the complete solution set for \(x \leq 4\).

Number lines are handy not just for simple inequalities but also for understanding complex relationships between numbers, making them a fundamental tool for visual learners.