Problem 42 - 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. \(\frac{x}{4}<-1\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -4\). This is represented by an open circle at -4 on a number line, with the area to the left shaded.

Step 1: Identify the coefficient

The term we are trying to isolate, \(x\), has a coefficient of 1/4. Therefore, to isolate \(x\), we will need to eliminate this fraction by multiplying it with the reciprocal.

Step 2: Multiply both sides of the inequality by 4

We multiply both sides of the inequality by 4, which is the reciprocal of 1/4. This is done in order to remove the fractions. This results in \(x < -4\).

Step 3: Graphing the inequality

To graph this inequality on a number line, we draw a line where the point representing -4 will be colored. Because the inequality is 'less than' and not 'equal to', we use an open circle to denote -4. We shade in the area of the number line to the left of -4, showing that all of these values are solutions to the inequality.

Key Concepts

Solving Inequalities

To solve inequalities using the multiplication property, follow similar steps as when solving equations, but keep in mind that inequalities involve comparison between values. The multiplication property of inequality states that when you multiply or divide both sides of an inequality by a positive number, the inequality remains the same. Let's go through the steps for solving the inequality \( \frac{x}{4} < -1 \):

• First, identify any fractions or coefficients attached to the variable \(x\). In our case, \(x\) has been divided by 4.
• To isolate \(x\), multiply both sides of the inequality by 4, the reciprocal of \(\frac{1}{4}\).
• This action will cancel out the fraction, leaving you with \(x < -4\).

Always remember: if you multiply or divide by a negative number, you must reverse the inequality sign. This rule ensures that the truth of the inequality remains intact, preserving the logical relationships between the numbers.

Graphing Inequalities

Graphing inequalities helps visualize the range of potential solutions on a number line. Here is how you can do it for \(x < -4\):

• Begin by drawing a horizontal line. This line will represent your number line.
• Next, identify the point \(-4\) on this line. Since the inequality is "less than" and does not include \(-4\) itself, you mark this point with an open circle.
• The open circle means that \(-4\) is not part of the solution set, indicating the solution range starts just to the left of \(-4\).
• Finally, shade the number line to the left of \(-4\). The shaded part indicates all the numbers that are solutions for \(x < -4\).

"Less than" and "greater than" inequalities will use open circles, while "less than or equal to" and "greater than or equal to" will use solid circles to include the point itself in the solution.

Number Line Representation

A number line is a visual tool that represents numbers as equally spaced points along a straight line. Utilizing the number line when solving inequalities like \(x < -4\) helps in understanding which numbers fit a given condition. Here's how to perform a number line representation effectively:

• First, label your number line with a reasonable scale that covers the focus number, in this case, \(-4\), and extends to the left.
• Mark \(-4\) with an open circle to indicate that it is not an inclusive part of the solution set since the inequality is strict (less than).
• Color or shade the line going left from \(-4\). This shading represents that all of these numbers satisfy \(x < -4\).

Number lines simplify visual interpretations of inequalities and can help to communicate solutions clearly, making it easier to grasp their meaning and range.