91Ó°ÊÓ

To solve an inequality means to find all its solutions, which are the values that make the inequality true. Just like solving equations, you'll often need to perform operations such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. However, remember that if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. In the exercise provided, the first inequality is solved by dividing both sides by 7. This isolates the variable on one side, resulting in a simpler inequality of the form \( x > a \) where \( a \) is the solution to the first part.

Similarly, the second inequality is solved by dividing both sides by 5, giving us \( x \leq b \), with \( b \) representing the solution to the second part. To improve understanding, consider running through a few example numbers to check if they satisfy the inequality, reinforcing the concept that the solutions are all the numbers that make the inequality true.

Graphing is a visual representation of all the possible solutions to an inequality on a number line or a coordinate plane. For simple, one-variable inequalities, a number line is sufficient. To graph a compound inequality, each individual inequality is graphed, and then the common solution area is identified.

In our exercise, after solving the individual inequalities, you plot them on the same number line. Use an open circle to indicate that a number is not part of the solution for strict inequalities like 'greater than' (>) and a closed or filled circle for 'less than or equal to' (\(\leq\)) when the number is included in the solution set. The intersection of the two regions, where they overlap, represents the solution to the compound inequality. As an advice for further understanding, practice graphing inequalities with different inequality symbols to recognize the patterns how each symbol is represented on the graph.

Inequality symbols specify the relationship between two values. The most common symbols are 'greater than' (\(>\)), 'less than' (\(<\)), 'greater than or equal to' (\(\geq\)), and 'less than or equal to' (\(\leq\)). In a strict inequality, like \(>\) or \(<\), the value on one side of the symbol is not included as a solution; hence the open circle when graphing. In contrast, non-strict inequalities, namely \(\geq\) or \(\leq\), include the value as a possible solution, hence the filled circle on a graph. For students to grasp this concept effectively, it is recommended to compare and contrast pairs of similar inequalities, like \(x > 1\) and \(x \geq 1\), to see how adding the 'equal to' portion affects the solution set and its graphical representation.