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Solving inequalities involves finding the range of values that satisfy a given inequality statement. Much like solving equations, the goal is to isolate the variable. However, inequalities have some unique properties, particularly when it comes to direction. If you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip.
This exercise starts with the inequality \(4c + 4 > 32\). The first step is to isolate the term with the variable, in this case, \(4c\). To do this, subtract 4 from both sides, resulting in \(4c > 28\).
This is similar to solving an equation, where you perform the same operation on both sides. After isolating the variable term, you solve for \(c\) by dividing both sides by 4, leading to \(c > 7\). Remember, when dividing by a positive number, the direction of the inequality remains the same.

Graphing inequalities on a number line provides a visual representation of the solutions. For the inequality \(c > 7\), start by drawing a number line. Include essential points, such as 6, 7, and 8.
An open circle is placed at the number 7 on the number line. This open circle indicates that 7 itself is not included in the solution. If the inequality were \(c \geq 7\), a closed circle would be used to show that 7 is included.
To complete the graph, shade the portion of the number line that represents all numbers greater than 7. This shading to the right of the open circle indicates all possible values of \(c\) that satisfy the inequality \(c > 7\). Utilizing this method to represent inequality solutions enhances comprehension and clarity.

Number lines are a simple yet powerful tool for visualizing mathematical concepts, especially inequalities. They help in understanding which numbers satisfy an inequality.
To create a number line, draw a straight horizontal line and mark evenly spaced points that represent numbers. Choose enough points to clearly demonstrate the inequality solution. For example, in illustrating \(c > 7\), include 6, 7, and 8.

• Place numbers in increasing order from left to right.
• Use open or closed circles to denote whether endpoints are included.
• Shade regions to represent all values that satisfy the inequality.

Number lines not only aid in solving and graphing inequalities but also facilitate understanding of other numerical concepts, such as measure and estimations, making them an invaluable learning device.