Problem 32 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$-5 x \leq 30$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq -6\). It is represented by the interval notation \([-6, +\infty)\).

Step 1: Solve the inequality

To solve the inequality \(-5x \leq 30\), divide both sides by -5. Remember, when we divide or multiply an inequality by a negative number, the inequality symbol changes direction. So, \(x \geq -6\) is the solution to the inequality.

Step 2: Graph the solution on a number line

To graph \(x \geq -6\), plot a closed circle at -6, because the solution includes -6. Then, draw a line extending to the right, because x is greater than or equal to -6.

Step 3: Express the solution set in interval notation

The interval notation for \(x \geq -6\) is \([-6, +\infty)\). The brackets '[ ]' are used to include -6 in the solution set, and the parentheses '( )' are used for +\infty, since infinity is never included.

Key Concepts

Interval Notation

Interval notation is a way of representing a range of values, often used to describe the solution set of an inequality. In interval notation, we use brackets and parentheses to show whether endpoints are included or excluded.
For example:

• Square brackets [ ] indicate that an endpoint is included in the set.
• Parentheses ( ) indicate that an endpoint is excluded.

In the inequality solution given as \(x \geq -6\), the interval notation is \([-6, +\infty)\). This tells us that -6 is included in the interval, while infinity, being a concept rather than a number, cannot be included.

Graphing on a Number Line

Graphing solutions on a number line is a visual way to represent the range of values that satisfy an inequality. It helps in understanding which numbers are part of the solution set and their positioning relative to each other.
When graphing \(x \geq -6\), start by placing a closed circle at -6. The closed circle indicates that -6 itself is included in the solution set. Then, draw a line or arrow extending to the right from -6. This line represents all the numbers greater than -6 without end, illustrating the infinite nature of the set.

Solving Inequalities

Solving inequalities involves finding a range of values for the variable that makes the inequality true. The steps are similar to solving equations, but there are special rules to follow:

• Perform the same operation on both sides of the inequality.
• When multiplying or dividing by a negative number, flip the inequality symbol.

In the exercise example, we divide both sides of \(-5x \leq 30\) by -5. This operation flips the inequality: \(x \geq -6\). Remembering this rule is crucial to correctly solving inequalities.

Inequality Symbols

Inequality symbols are mathematical signs that show the relationship between values in inequalities. The key symbols are:

• \(<\) for 'less than'
• \(>\) for 'greater than'
• \(\leq\) for 'less than or equal to'
• \(\geq\) for 'greater than or equal to'

In our exercise, \(\leq\) indicates 'less than or equal to', which changes to \(\geq\) on solving the inequality due to division by a negative number. Understanding these symbols is essential for interpreting and solving inequalities effectively.