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Understanding how to visualize inequalities is crucial for grasping the range of solutions possible. To graph an inequality like \( x < 3 \), you start with a number line, which is a straight, horizontal line that has numbers placed at intervals. Every point on this line represents a real number. To graph \( x < 3 \), you would locate the number 3 on the number line and make an open circle around it to indicate that 3 itself is not included in the solution set.

Next, considering the inequality signifies all numbers less than 3, you would shade the entire portion of the number line that extends to the left of the number 3. The shaded area shows all numbers that make the inequality true, offering a visual representation to better understand the concept. It's important to remember that the open circle and the direction of the shading depends on the inequality sign used.

The number line is a fundamental tool for illustrating mathematical concepts, especially inequalities. It helps students see the set of possible solutions in a straightforward manner. For an inequality like \( x < 3 \), the solution set consists of all values of \( x \) that are less than 3. On the number line, this is represented by shading the section left of the 3 and not including the number 3 itself, as evidenced by the open circle.

Interpreting Number Line Graphs

When looking at a graphed inequality on a number line, always check the type of circle used at a boundary point. An open circle means that the boundary number is not part of the solution set, while a closed circle would mean it is included. Also, the direction of the shading indicates whether the inequality is less than (leftward shading) or greater than (rightward shading). This visual aid supports the understanding of range and limits within mathematical solutions.

Manipulating inequalities is similar to solving equations, but with special attention to the direction of the inequality sign. In the given exercise, the inequality \(14 - 3x > 5\) is simplified by subtracting 14 from both sides to get \(-3x > -9\). To solve for \(x\), you divide by the coefficient of \(x\), which is -3, resulting in \( x < 3 \).

Reversing the Inequality

A key step in inequality manipulation is to reverse the direction of the inequality sign when multiplying or dividing by a negative number. This principle is vital to maintain the truth of the inequality, as the nature of the order reverses when both sides are affected by a negative multiplier or divisor. Therefore, understanding and applying this rule of inequality manipulation ensures that the solution to the inequality maintains its integrity and logic.