Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. $$-2 y \leq \frac{1}{2}$$

Step-by-Step Solution

Answer

The solution to the inequality -2y ≤ 1/2 is y ≥ -1/4. For the graph, shade the number line to the right of -1/4 including -1/4 itself (depicted by a closed circle).

Step 1: Identification

Recognize the given inequality -2y ≤ 1/2. Y is being multiplied by -2 hence, we need to get rid of -2 to isolate y.

Step 2: Applying the multiplication property of inequality

Divide both sides of the inequality by -2 to isolate y. Hence, the inequality becomes y ≥ -1/4. Remember that when you divide by negative number you have to flip the sign of inequality

Step 3: Graphing solution on a number line

On a number line, y values range from negative infinity to positive infinity. The inequality y ≥ -1/4 means that y is any number greater than or equals to -1/4. This can be represented by a closed circle at -1/4 (since -1/4 is included) and shading to the right (towards +inf).

Key Concepts

Solving Inequalities

When dealing with inequalities, we aim to find all the values of the variable that make the inequality true. Solving inequalities is similar to solving equations, with the added twist that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Take the exercise \( -2y \leq \frac{1}{2} \). To isolate \(y\), we divide both sides by \( -2 \), flipping the inequality to \( y \geq -\frac{1}{4} \) because we divided by a negative number. This step is critical; failing to flip the sign when dividing or multiplying by a negative would yield the incorrect solution set.

Once \(y\) is alone on one side, we can interpret the inequality: \(y\) can be any number greater than or equal to \( -\frac{1}{4} \). It's essential to read the inequality correctly, as the direction of the sign has deep implications for the set of solutions.

Number Line Graphing

Graphing inequalities on a number line helps visualize the set of solutions. It's a simple yet powerful tool in understanding the range of values that satisfy the inequality.

For the given solution \(y \geq -\frac{1}{4}\), start by drawing a horizontal line to represent the number line. Then, locate the value of \( -\frac{1}{4} \) on this line. A closed circle is placed at \( -\frac{1}{4} \) to indicate that this number is part of the solution (the 'equal to' part of 'greater than or equal to'). From this point, shade the line to the right, signifying all numbers greater than \( -\frac{1}{4} \) are solutions.

Remember, an open circle would be used for 'greater than' but not 'equal to' (when the inequality does not include the boundary). Number line graphs are intuitive: they make dealing with positive and negative values, as well as ranges of solutions, more understandable.

Inequality Properties

Inequalities hold specific properties that guide how they can be manipulated during solving processes. The multiplication property involved in the given exercise is just one of these. Understanding all the properties of inequalities is crucial for correctly solving them.

Some important inequality properties include:

• Addition/Subtraction Property: You can add or subtract the same number to both sides of an inequality without changing its direction.
• Multiplication/Division Property: Multiplying or dividing both sides by a positive number does not change the direction of the inequality.
• Division by a Negative: As demonstrated in the exercise, dividing both sides by a negative number reverses the inequality's direction.
• Transitive Property: If \(a \leq b\) and \(b \leq c\), then \(a \leq c\).

It's also worth noting that treating an inequality like an equation (except when multiplying or dividing by a negative number) generally leads to the correct solution set. These properties ensure that inequalities remain balanced as they are solved, similar to the concept of maintaining an equation's balance.