Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. $$-3 x<15$$

Step-by-Step Solution

Answer

The solution to the inequality \(-3x < 15\) is \(x > -5\).

Step 1: Divide each side of the inequality by -3

Applying the multiplication property of inequality, each side of the equation \(-3x < 15\) is divided by -3. Remember, whenever an inequality is multiplied or divided by a negative number, the inequality sign flips. Therefore, the inequality becomes \(x > -5\).

Step 2: Graph the solution on a number line

Next, represent the solution \(x > -5\) on a number line. Start by drawing a line and marking off the segment representing -5, which is the point of reference. The solution to this inequality includes all numbers greater than -5, so on the number line, you need to draw an open circle at -5 (indicating that -5 is not part of the solution set) and an arrow extending to the right (representing all numbers greater than -5).

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is key to solving inequalities like \(-3x < 15\). It tells us that when we multiply or divide both sides of an inequality by the same number, the inequality remains true as long as we follow one critical rule. If that number is negative, we must flip the inequality sign.
For example, with \(-3x < 15\), divide both sides by -3. The inequality becomes \(x > -5\) because dividing by a negative flips the "less than" sign to "greater than." This rule can trip up students, so it's important to be mindful when working with negative numbers in inequalities.

• Always flip the inequality sign when multiplying or dividing by a negative.
• This rule ensures that the order of numbers remains consistent.

Solving Inequalities

Solving inequalities involves finding a set of values that make the inequality true. In our problem, \(-3x < 15\), we solve by isolating \(x\). Here's how:
First, divide both sides by -3, remembering to flip the inequality sign due to the negative divisor. This isolates \(x\) and results in \(x > -5\).

• Start by simplifying both sides where possible.
• If you multiply or divide by a negative, flip the sign for a correct solution.
• Check your solution by substituting a number from the solution set to see if it satisfies the original inequality.

This process helps you understand the range of values that make the inequality true.

Graphing Inequalities

Once you've solved an inequality like \(x > -5\), representing the solution visually on a number line offers clarity. Here's a simple guide:
Draw a number line and mark the critical point, in this case, -5. Because our inequality is "greater than" but not "equal to," we use an open circle to indicate that -5 itself isn't part of the solution set.
Then, draw an arrow extending to the right from -5. This arrow shows that all numbers greater than -5 satisfy the inequality.

• An open circle means the number is not included (\(<, >)\).
• A closed circle is used when the number is included (\(\leq, \geq)\).
• The direction of the arrow indicates which side of the number is included in the solution.

This visual step is crucial for a solid understanding of the solution set and helps in verifying your work.