Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$2 y-5<5 y-11$$

Step-by-Step Solution

Answer

The solution to the inequality is \(y > 2\). On a number line, this is represented by an open circle at '2' and the line extending to the right from '2'.

Step 1: Isolate 'y' terms on one side of the inequality

Begin by arranging the inequality to have 'y' terms on one side. Subtract '2y' from both sides of the inequality to get '2y - 2y - 5 < 5y - 2y -11' which simplifies to '-5 < 3y - 11'

Step 2: Isolate 'y'

Next, aim to isolate 'y' to one side by adding '11' to both sides of the inequality to eliminate '-11' from the right. This gives: '-5 + 11 < 3y - 11 + 11', simplifying to '6 < 3y'.

Step 3: Solve for 'y'

Now, divide both sides of the inequality by '3' to isolate 'y'. This gives '6 / 3 < 3y / 3' which simplifies to '2 < y'.

Step 4: Graph the solution

On the number line, 'y' is greater than '2' so the graph will be an open circle at '2' and all the points to the right of '2', since 'y' can take any value greater than '2'. Points to the left of '2' are not included as 'y' cannot be less than '2'.

Key Concepts

Addition Property of Inequality

The addition property of inequality is a fundamental concept in mathematics that helps us maintain the relationship between two sides of an inequality even when we add the same number to both sides. To comprehend this, let's consider the original inequality: \(2y - 5 < 5y - 11\).

Our objective is to rearrange this inequality so that all terms containing \(y\) are on one side, and the constant terms are on the other. By adding \(5\) to both sides, we eliminate the constant on the left:

• Initial inequality: \(2y - 5 < 5y - 11\)
• After adding \(5\): \(2y < 5y - 6\)

This simple addition step helps clarify and continues to hold true due to the properties of inequalities.
Keep in mind that if the inequality signs were reversed (for instance, \(>\) or \(\geq\)), the method of solving remains consistent.

Multiplication Property of Inequality

The multiplication property of inequality allows us to solve inequalities where the term we want to isolate is being multiplied.
If you multiply both sides of an inequality by a positive number, the inequality direction stays the same.
However, when you multiply both sides by a negative number, you need to reverse the inequality sign.

For the inequality \(6 < 3y\), we divide both sides by \(3\), a positive number, preserving the inequality direction:

• \(6 \div 3 < 3y \div 3\)
• This simplifies to: \(2 < y\)

Such operations demonstrate how multiplicative actions isolate variables while maintaining the original inequality relationship. Remember, the rule to flip the inequality sign only applies when multiplying or dividing by negative numbers.

Graphing Inequalities

Graphing is a visual representation of the solutions for inequalities, making it easier to understand and interpret. In the inequality \(2 < y\), we indicate that \(y\) can be any value greater than 2.

To graph this on a number line:

• Draw a number line.
• Place an open circle on 2, showing that 2 is not included in the solution set.
• Draw a line extending to the right from that open circle, indicating all numbers greater than 2 are solutions.

Using a number line simplifies recognizing the range of possible solutions at a glance.

Solving Inequalities

Solving inequalities involves arranging and manipulating equations to find the set of solutions. The goal is to isolate the variable of interest on one side.

Consider again our exercise where we simplified \(-5 < 3y - 11\) to \(6 < 3y\) and then solved it to reach \(2 < y\). This entire process involves:

• Rearranging terms using addition or subtraction.
• Isolating the variable using multiplication or division.

Every step requires careful attention to maintain the inequality's truth. In contrast to equations, inequalities suggest a relationship of greater or less than, rather than equality, affecting how we interpret and graph our solutions.