Problem 71 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(3(2 y-1)<9\)

Step-by-Step Solution

Answer

The solution to the inequality is \(y<2\). This can be indicated on a number line by placing an open circle at \(2\) and shading the line to the left of \(2\).

Step 1: Distribute Multiplication

Multiply \(3\) by \(2 y - 1\), which gives \(6 y - 3 < 9\)

Step 2: Add 3 to both sides

After applying the addition property of inequality by adding \(3\) to both sides of the inequality, the inequality simplifies to \(6 y < 12\)

Step 3: Divide by 6

Applying the multiplication property of inequality which involves dividing both sides by \(6\), the inequality further simplifies to \(y < 2\).

Step 4: Graph the solution on a Number Line

Draw a number line, place a open circle on \(2\) and darken the line to the left of \(2\) to represent all numbers less than \(2\)

Key Concepts

Addition Property of Inequality

The addition property of inequality is an essential tool in solving inequality problems. As seen in our exercise, by adding the same number to both sides of an inequality, the inequality remains true. For instance, given the inequality \(6y - 3 < 9\), we can add \(3\) to both sides to obtain \(6y < 12\). This property is fundamental because it allows us to isolate the variable, making the inequality easier to solve.

Exercise Improvement Advice: To ensure the concept is clear, remember that when you add a positive number to both sides, the direction of the inequality does not change. It's a straightforward maneuver akin to balancing scales; you're keeping the inequality balanced by doing the same action to both sides.

Multiplication Property of Inequality

Solving inequalities often requires manipulating them through multiplication or division. This is where the multiplication property of inequality comes into play. It states that if you multiply or divide both sides of an inequality by the same positive number, the inequality will stay in the same direction. However, and critically, if the number is negative, the inequality sign reverses. In our exercise, dividing both sides of the inequality by \(6\) keeps the inequality sign unchanged, resulting in \(y < 2\).

Exercise Improvement Advice: Students may commit errors when multiplying or dividing by negative numbers. To avoid this, emphasize that the rule of sign flipping applies exclusively when working with negative multipliers or divisors.

Graphing Inequalities on a Number Line

Graphing inequalities on a number line provides a visual representation which helps to better understand the range of possible solutions. In our example, once we've simplified the inequality to \(y < 2\), we illustrate the solution on a number line. Here, an open circle is placed on the number \(2\) to signify that \(2\) is not included in the solution while the darkened line to the left represents all values of \(y\) that make the inequality true - values less than \(2\).

Exercise Improvement Advice: For clarity in graphing, using an open circle for '<' or '>' and a closed or filled-in circle for '\(\leq\)' or '\(\geq\)' can help differentiate between those values included in the solution set and those that are not. This demarcation is critical for correctly interpreting inequality solutions.

Inequality Solutions

The solution to an inequality is the set of all values that satisfy the inequality. After using both the addition and multiplication properties of inequality, like we did in the exercise to get from \(3(2y - 1) < 9\) to \(y < 2\), we find that any number less than \(2\) is a solution. It's a broader concept as compared to equations, where we often find a single solution. Inequalities tell us about a range of possible solutions, each of which makes the original inequality true when substituted for the variable.

Exercise Improvement Advice: To solidify understanding, students can substitute values back into the original inequality to verify that they indeed satisfy the inequality. Moreover, practice with a variety of inequalities helps students become comfortable with the process and the logic behind finding solutions for these mathematical statements.