Problem 46 - 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. $$7 x \geq-56$$

Step-by-Step Solution

Answer

The solution to the inequality \(7x \geq -56\) is \(x \geq -8\).

Step 1: Solving the Inequality

To solve the inequality \(7x \geq -56\), the multiplication property of equality is applied which states that one can multiply or divide both sides of an inequality by the same non-zero number and the inequality will still hold. Here, \(x\) will be isolated on one side by dividing both sides of inequality by \(7\). Hence, \(x \geq -56/7\). After simplifying, \(x \geq -8\). Therefore, the solution of the inequality is \(x \geq -8\).

Step 2: Graphing The Solution Set On A Number Line

To graph the solution set on a number line, firstly, a straight horizontal line is drawn which is known as a number line. Then, -8 is marked on this number line. Since the inequality is \(x \geq -8\), an filled dot is drawn at -8 and an arrow is drawn to the right of -8 which shows all the numbers greater than or equal to -8 are solutions to the inequality. Thus, the solution set is graphed on a number line.

Key Concepts

Multiplication Property of Inequality

Understanding the multiplication property of inequality is crucial when solving inequalities. This property tells us that if we multiply or divide both sides of an inequality by the same positive number, the direction of the inequality does not change. For example, if we have an inequality like \( a > b \), and we multiply both sides by a positive number \( c \), the inequality \( ac > bc \) still holds true. Similarly, when we divide both sides by a positive number, the inequality remains in the same direction.

However, a pivotal point to remember is that if we multiply or divide by a negative number, the inequality flips. That means if \( a > b \) is multiplied by a negative number \( -c \), we get \( -ac < -bc \). With the exercise at hand, \( 7x \geq -56 \), we divide both sides by a positive 7 to isolate \( x \), resulting in \( x \geq -8 \) without changing the inequality's direction.

Number Line Graphing

Graphing on a number line is a visual way to represent inequalities. A number line is a straight, horizontal line with numbers placed at equal intervals. When graphing inequalities, it's crucial to use an open dot for inequalities like \( x > a \) or \( x < a \), which indicate that the number 'a' is not part of the solution. Conversely, for inequalities such as \( x \geq a \) or \( x \leq a \), a filled dot is used, signalling that 'a' is included in the set of solutions.

In our exercise, since we have \( x \geq -8 \), we mark -8 on the number line with a filled dot to include it in the solution set. Then, an arrow is drawn to the right, representing all the numbers that are greater than or equal to -8. This graphical representation helps students visually understand the range of possible solutions to the inequality.

Inequality Solutions

The solutions to an inequality consist of all the values that make the inequality true. For the inequality \( x \geq -8 \), the solution is not just a single number but a range of numbers. These solutions are sometimes referred to as the 'solution set' and can be expressed in various forms -- as an inequality (like the original \( x \geq -8 \)), in set-builder notation, using a number line, or in interval notation which would be \( [-8, +\infty) \) in this case.

It's important to understand that the solution to an inequality is explained in terms of the range of values that satisfy the inequality rather than a precise answer. Practicing with different representations helps students grasp the breadth of inequality solutions better and how to accurately convey the set of all possible answers that fit the given inequality.