91Ó°ÊÓ

Understanding how to solve inequalities is a fundamental skill in algebra. Much like solving equations, solving inequalities requires isolating the variable on one side. The solution to an inequality shows the range of values that the variable can take.

For instance, let's look at the compound inequality from the exercise: solving each inequality involves similar steps to solving regular equations. For the inequality \(6c \text{≤} 18\), the first step is dividing both sides by 6, yielding \(c \text{≤} 3\). This indicates that \(c\) could be 3 or any number less.

However, inequalities have a twist when it involves dividing by a negative value. For the inequality \(-5c \text{≤} 15\), dividing by -5 requires us to reverse the inequality sign, leading to \(c \text{≥} -3\). This highlights that whenever we multiply or divide an inequality by a negative number, we must flip the inequality sign to maintain the truth of the statement. This step is crucial and often a source of errors if overlooked. In the context of our compound inequality, we have two valid ranges for \(c\), one for each part of the compound inequality.

Graphing inequalities aids in visualizing the range of possible solutions. A number line is traditionally used to represent the solutions. Unlike equations, where we have a distinct point, inequalities cover a range and are depicted with shading.

For our exercise, we graph each inequality separately before combining them. The inequality \(c \text{≤} 3\) is represented by shading everything to the left of 3 including the number 3 itself. It's marked by a filled circle on 3 to indicate inclusion. Similarly, \(c \text{≥} -3\) is graphed by shading to the right of -3, which also includes -3, shown with another filled circle.

Since the original problem states 'or', we include both sets of shaded regions. The filled circles demonstrate that -3 and 3 are also part of the solution. Therefore, the complete solution on a number line extends from left to right, covering all numbers, with no gap between -3 and 3.

Inequality notation conveys the relationship between values concisely and accurately. There are several types of inequality symbols: '<' means 'less than,' '>' means 'greater than,' '≤' stands for 'less than or equal to,' and '≥' for 'greater than or equal to.' The compound inequality from our exercise uses '≤' and '≥', illustrating that the values we find for \(c\) can be exactly at the boundary numbers or beyond them in the direction the inequality indicates.

When dealing with compound inequalities involving 'or', such as in our exercise, the solution set is a union of the solutions of each inequality. It represents all the values that satisfy either of the inequalities. On the other hand, compound inequalities that use 'and' would only include values that satisfy both inequalities simultaneously, which often results in a smaller range of solutions. Understanding and applying the correct inequality notation is essential for accurately representing and communicating solutions.