Problem 12 Graph the solution set, and writ... [FREE SOLUTION] (2024)

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Inequalities are mathematical expressions that show the relationship between two values where one is not necessarily equal to the other. They are indicated by symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

For example, in the inequality \(x - 3 \geq 7\), we are dealing with a 'greater than or equal to' relationship. Solving inequalities often involves similar steps to solving equations, but we must remember that if we multiply or divide by a negative number, we flip the inequality sign.

When we solve inequalities, we often need to express our solution in interval notation. Interval notation is a way of writing sets of numbers as intervals. It includes two numbers, the start and the end of the range, within which all included numbers fall.

For instance, the solution \(x \geq 10\) in interval notation is written as \([10, \infty)\). Here:

• The square bracket \([\) indicates that 10 is included in the interval (closed interval).
• The parenthesis \(()\) next to infinity indicates that infinity is not included (open interval).

This tells us that the solution includes all numbers from 10 to infinity, including 10 itself.

Graphing inequalities on a number line helps visualize the solution set. Follow these steps to graph inequalities:

• First, draw a horizontal line representing all possible numbers.
• Next, mark the specific value (like 10 in our exercise) on the line. For \(x \geq 10\), place a closed circle at 10 to show that this point is included in the solution set.
• Finally, shade the line to the right of 10 to represent all values greater than or equal to 10.

This visually shows that any point on or to the right of 10 satisfies the inequality.

A number line is a straight, horizontal line with numbers placed at intervals. It helps in understanding numerical values and their relationships easily.

To use a number line for inequalities:

• Identify the critical value or values of the inequality.
• Use open circles for values that are not included (like with < or >).
• Use closed circles for values that are included (like with ≤ or ≥).
• Shade the part of the line that represents the solution set.

For instance, in \(x \geq 10\), you'd place a closed circle on 10 and shade everything to the right to infinity, visualizing all numbers greater than or equal to 10.