Solve each inequality. Write each solution set in interval notation. Example 1 and Example...

Solve each inequality. Write each solution set in interval notation. Example 1 and Example 2.

EXAMPLE

Solving a Linear Inequality

Solve −3x + 5 > −7.

SOLUTION

Thus, the original inequality − 3x + 5 > −7 is satisfied by any real number less than 4. The solution set can be written {x |x < 4}. A graph of the solution set is shown in Figure, where the parenthesis is used to show that 4 itself does not belong to the solution set. Note that testing values from the solution set in the original inequality will produce true statements, while testing values outside the solution set produces false statements.

Figure

The solution set of the inequality,

is an example of an interval. We use a simplified notation, called interval notation, to write intervals. With this notation, we write the interval as

The symbol − ∞ does not represent an actual number. Rather it is used to show that the interval includes all real numbers less than 4. The interval ( − ∞, 4) is an example of an open interval, since the endpoint, 4, is not part of the interval. A closed interval includes both endpoints. A square bracket is used to show that a number is part of the graph, and a parenthesis is used to indicate that a number is not part of the graph.

EXAMPLE

Solving a Linear Inequality

Solve 4 − 3x ≤ 7 + 2x. Give the solution set in interval notation.

SOLUTION

In interval notation the solution set is . See Figure for the graph.

Figure

−2x + 8 ≤ 16