Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. $$-12 y+17>20-13 y$$

Step-by-Step Solution

Answer

\(y > 3\)

Step 1: Simplify the inequality by combining like terms

First, we need to bring terms with 'y' together on one side for easy comparison. \( -12y + 13y > 20 - 17 \)

Step 2: Solve the simplified inequality

The inequality becomes \( y > 3 \) after the subtraction operation.

Step 3: Graph the solution on the number line

We then graph the solution on a number line, marking '3' as a boundary point. The solution will be greater than 3 i.e all the numbers to the right of '3' will be included.

Key Concepts

Addition Property of Inequality

The addition property of inequality is an essential concept when solving inequalities. It allows us to add or subtract the same number from both sides of an inequality without altering the inequality's truth. This property is similar to the addition property in equations but requires attention to direction when dealing with inequalities.
For instance, consider the inequality \(-12y + 17 > 20 - 13y \). To solve this using the addition property, we can move terms with 'y' to one side by adding \(13y\) to both sides, giving us \(-12y + 13y + 17 > 20\). This simplifies the inequality to \(y + 17 > 20\).
Once simplified, you can further isolate 'y' by using additional subtraction or addition actions based on what remains, like subtracting \(17\) to eventually resolve that \(y > 3\).
This property maintains the inequality relationship as long as the operations involve either addition or subtraction. However, stay cautious about multiplying or dividing by negative numbers, which can reverse inequality signs.

Graphing Inequalities

Graphing inequalities helps visualize the range of solutions on a number line. When graphing, concentrate on marking the boundary point accurately, which is the solution where the inequality becomes an equation.
In our case, the simplified inequality \(y > 3\) tells us that '3' is the boundary point. This point differentiates between where 'y' values satisfy the inequality and where they do not. To represent \(y > 3\), '3' becomes our starting reference on the number line.

• Use an open circle at '3' to specify that it is not included in the solution set.
• Draw an arrow extending to the right from '3' indicating all numbers greater than '3' satisfy the inequality.

Ensure that the arrow direction and the nature of the boundary point (open or closed circle) correspond precisely to the inequality type, letting you communicate the solution set accurately.

Number Line Representation

A number line representation provides a clear visual of all possible solutions to an inequality. It is a simple tool that effectively displays the "set" of numbers complying with the inequality rule.
When dealing with an inequality like \(y > 3\), here's how to represent it on a number line:

• Draw a horizontal line and evenly space numbers around your boundary point, '3'.
• Place an open circle at '3' because \(>\) signifies that 'y' does not equal '3'.
• Shade or extend a line with an arrow from '3' to the right, illustrating \(y > 3\).
• Remember, a closed circle would mean an inclusive boundary, which is not the case here.

This visual representation makes it easier to see which numbers form part of the solution, aiding in understanding and communication of mathematical inequalities.