91Ó°ÊÓ

When working with inequalities, we often come across problems that involve more than one inequality statement. These are called compound inequalities, and they require us to consider two conditions simultaneously. A compound inequality like \( -2 < x \leq 5 \) combines two relationships: the value of \( x \) is greater than \( -2 \) and at the same time, it is less than or equal to \( 5 \). This dual nature distinguishes compound inequalities from simple inequalities, which contain only one relationship.

To solve a compound inequality, we first solve each inequality separately. For example, \( x > -2 \) and \( x \leq 5 \) are our two conditions. Then we find the overlap between the solutions of each inequality, which is the range of \( x \) values that satisfy both conditions at the same time. If there is no overlap, it means that there is no common solution. But in cases where there is an overlap, like in the exercise given, we can graph this on a number line to visualize the solution.

The number line is a simple yet powerful tool to represent inequalities. To use a number line, you draw a horizontal line, mark points for critical values from the inequality, and use different types of dots or circles to show whether those values are included or excluded in the solution set.

In our current exercise, the number line allows us to represent the compound inequality by clearly showing that the solution for \( x \) falls between two numbers, \( -2 \) and \( 5 \). We place a hollow circle above \( -2 \) to denote that it is not part of the solution set, and a filled circle above \( 5 \) to signify inclusion. Connecting these two points with a line or bar expresses all the numbers between them, forming a continuous range of solutions.

The solution to an inequality tells us the range or set of numbers that make the inequality true. For the single inequality \( x > -2 \) the solution is all values of \( x \) that are greater than \( -2 \). However, because we're dealing with a compound inequality, we must also consider the second part: \( x \leq 5 \). This means we want all values of \( x \) that are simultaneously greater than \( -2 \) and less than or equal to \( 5 \).

The combination of these solutions can be infinite, as there are infinitely many numbers between \( -2 \) and \( 5 \), including fractions and irrational numbers. To illustrate this on a number line can be tricky, as we can't pinpoint every possible number. Instead, we use the line between the two marked points to symbolize all these numbers as part of the solution set, thereby communicating a potentially infinite and continuous range of answers.