6-6 Inequalities

Property

To write an inequality from a verbal phrase, you match keywords to their corresponding inequality symbols.

• Less than ($<$): "is less than", "is smaller than"
• Greater than ($>$): "is greater than", "is more than"
• Less than or equal to ($\leq$): "is at most", "is no more than", "maximum"
• Greater than or equal to ($\geq$): "is at least", "is no less than", "minimum"

Property

To show the solution set of an inequality on a number line:

• For inequalities with $\leq$ or $\geq$, use a filled-in dot to show the endpoint is included in the solution.
• For inequalities with $<$ or $>$, use an open circle to show the endpoint is not included.
• Shade the region of the number line that represents all possible solutions, often with an arrow to show it continues infinitely.

Examples

• To graph $x < 2$, place an open circle at 2 and shade the number line to the left.
• The graph for $x \geq -1$ has a filled-in dot at -1 and shading to the right.
• The solution set for $x > 5$ is shown with an open circle at 5 and an arrow shading everything to the right.

Explanation

Think of the dot as a gate. A filled-in (closed) dot means the gate is part of your property (the solution). An open circle means it's just a post marking the boundary, but it isn't included.

Property

A solution to an inequality is any value that makes the statement true. To check, substitute the value for the variable and see if the resulting inequality holds.

Examples

• Is 6 a solution for $4y - 5 \ge 11$? Yes, because $19 \ge 11$ is true.
• Is 2 a solution for $4y - 5 \ge 11$? No, because $3 \ge 11$ is false.

Explanation

Think of it as trying a key in a lock. You plug the number in, and if the inequality “unlocks” by making a true statement, it's a solution! If not, that number isn't in the solution club.