Problem 47 - Introductory Algebra for College Students

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 is \(x > -5\). The solution on the number line is represented by an arrow pointing to the right starting from -5, excluding -5 itself.

Step 1: Division step

First of all, to isolate \(x\) in the given inequality \(-3x < 15\), divide each side of the equation by -3. Keeping in mind that when dividing an inequality by a negative number, the direction of the inequality sign flips, the inequality becomes: \(x > -5\).

Step 2: Drawing the Number line

To graph this inequality, draw a number line, mark 0, and negative numbers in order. Place an open circle at -5 since it is not included in the solution set (we have > sign, not \(\geq\)). Draw a line starting from -5, to the right, showing all the greater than -5 numbers.

Step 3: Represent the solution set

Everything to the right of -5 (excluding -5 itself) is included in the solution set. We demonstrate this by using an arrow pointing to the right, showing that \(x\) is greater than -5.

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is a fundamental rule used when solving inequalities. It is essential to remember that multiplying or dividing both sides of an inequality by a positive number keeps the inequality sign the same. However, when you multiply or divide by a negative number, the direction of the inequality sign must change. This rule ensures that the relationship between the quantities is preserved. In the exercise example, we have the inequality \(-3x < 15\), and to solve for \(x\), you divide both sides by \(-3\). Doing so flips the inequality sign, resulting in \(x > -5\). It’s a crucial step to remember when dealing with inequalities, as missing this change can lead to incorrect solutions! By practicing this rule, you will boost your confidence in solving inequalities.

Number Line Graphing

Number line graphing is a visual method used to represent inequality solutions. Here, it serves as a tool to show all possible values that satisfy an inequality. For instance, when the solution to an inequality is \(x > -5\), we need to demonstrate all numbers greater than \(-5\) on a number line. First, draw a horizontal line and mark zero and other numbers in their respective order.

• Identify the value \(-5\) and place an open circle on it. The open circle indicates that \(-5\) is not included in the solution set.
• Then draw a line with an arrow to the right, starting from \(-5\), showing all greater values.

The arrow signifies that the values continue infinitely beyond the explicitly marked numbers. This technique helps in visually grasping the range of solutions and is crucial for understanding inequalities better.

Algebra Problem Solving

Algebra problem solving involves finding the correct value or set of values that satisfy given equations or inequalities. It is a process that requires logical reasoning and application of algebraic rules. When solving inequalities, you usually perform steps similar to solving equations, such as isolating the variable. Always pay attention to any special rules, like those for the multiplication property of inequality, which affects the inequality sign. Breaking down the problem step by step, like dividing \(-3x < 15\) by \(-3\), helps in maintaining clarity. After solving, verify the solution by substituting values back into the original inequality. Graphing the solution on a number line is a great way to visually confirm your solution set. Approaching each problem methodically, with a clear strategy, can make algebra problem-solving manageable and even enjoyable!