Solve each compound inequality. Use graphs to show the...

Solve each compound inequality. Use graphs to show the solution set to each of the two given inequalities, as well as a third graph that shows the solution set of the compound inequality. Except for the empty set, express the solution set in interval notation.
$$x>3$ and $x>6$$

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Key Concepts

Compound Inequalities

Compound inequalities involve finding values that satisfy all of the given conditions simultaneously. In problems where multiple inequalities are joined by 'and', the solution is the intersection of the solution sets for each individual inequality, meaning only values that meet every condition are included.

Intersection of Solutions

The intersection of solutions refers to the set of values that simultaneously satisfy all the individual inequalities in a compound inequality. When one inequality is more restrictive than the other(s), the intersection is defined by the strongest condition.

Interval Notation

Interval notation is a concise way to represent sets of real numbers that satisfy an inequality. It uses parentheses and brackets to indicate open or closed endpoints, respectively. For example, the interval (6, ?) represents all numbers greater than 6.

Graphical Representation on a Number Line

Graphing inequalities on a number line visually depicts the set of solutions. An open circle is used to indicate that an endpoint is not included, and shading extends in the direction that satisfies the inequality. When graphing compound inequalities, it is helpful to graph each condition separately and then combine them to show their intersection.

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