Solve each inequality using a graphing utility. Graph each...

Solve each inequality using a graphing utility. Graph each of the three parts of the inequality separately in the same viewing rectangle. The solution set consists of all values of $x$ for which the graph of the linear function in the middle lies between the graphs of the constant functions on the left and the right. $$2 \leq 4-x \leq 7$$

Key Concepts

Compound Inequalities

A compound inequality involves two separate inequalities combined into one statement, where the solution must satisfy both conditions at the same time. Understanding compound inequalities is important because it requires finding the intersection of the solution sets of the individual inequalities, which, in many cases, can be visualized using a number line or a graph.

Linear Functions

A linear function is a function that can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept. In the context of inequalities, graphing the linear function helps to visualize the range of values x can take, especially when comparing it with constant functions representing boundary conditions.

Constant Functions

A constant function is a function of the form y = c, where c is a constant. In graphing, a constant function appears as a horizontal line. When using constant functions to bound another function in an inequality, their graphs help define the upper or lower limits, and the solution is the region where the other function’s graph is sandwiched between these lines.

Graphing Utility

Graphing utilities are powerful tools that allow users to visualize complex equations and inequalities by plotting functions on the same coordinate system. They assist in identifying the regions where the multiple conditions of a compound inequality are met, thereby giving a clear visual representation of the solution set as the intersection of the graphs.

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