Problem 76 - 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. \(7-2(x-4)<5(1-2 x)\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -1.25\). This is represented on the number line as a shading to the left of -1.25.

Step 1 Title: Distribute and Simplify

First, distribute terms inside the parentheses and simplify both sides. \(7 - 2(x - 4) < 5(1 - 2x)\) simplifies into: \(7 - 2x + 8 < 5 - 10x\) then to \(15 - 2x < 5 - 10x\)

Step 2 Title: Isolate the variable

Rearrange the inequality to bring terms with \(x\) together on one side and constant terms on the other side: add \(2x\) to each side and subtract \(5\) from each side: \(15 - 5 < 2x - 10x\). That simplifies into: \(10 < -8x\). Then divide by \(-8\) (and remember to flip the inequality as we are dividing by a negative number): \(10/-8 > x\). Therefore, \(x < -1.25\)

Step 3 Title: Represent on the number line

Draw a number line and mark the point -1.25 on it. Since the inequality is \(x < -1.25\), all points less than -1.25 make the inequality true, so shade the number line to the left of -1.25.

Key Concepts

Addition and Multiplication Properties

Understanding how to manipulate inequalities is fundamental to solving them. The addition and multiplication properties are two key tools at your disposal.

The addition property asserts that you can add or subtract the same value from both sides of an inequality without changing its direction. For instance, if you have an inequality like: \( a < b \), and you add \( c \) to both sides (assuming \( c \) is a real number), the inequality will remain true: \( a + c < b + c \). This property is useful for simplifying expressions and isolating variables.

Similarly, the multiplication property states that you can multiply or divide both sides of an inequality by the same positive value without altering the direction of the inequality. For example, if you have \( a < b \) and you multiply both sides by \( c \) (where \( c > 0 \) and is a real number), it stays: \( ac < bc \). It's crucial, however, to note that when you multiply or divide by a negative number, the inequality sign reverses. This is a common pitfall and an essential rule to remember: \( a < b \) divided by \( -c \) (where \( c < 0 \) and is a real number) results in the inequality \( a/c > b/c \). Applying these properties correctly allows for successful manipulation and solution of inequalities.

Inequality Graphing

Graphing an inequality on a number line provides a visual representation of all its solutions. In the exercise example, after isolating the variable, the solution was determined to be \( x < -1.25 \).

To graph this inequality, start by drawing a horizontal line, which symbolizes the number line. Mark the significant numbers, including the boundary defined by the inequality, in this case, '-1.25'. It's important to decide whether to use an open or closed circle: an open circle at '-1.25' shows that this number is not a solution to the inequality, while a closed circle would indicate it is included as a solution. Since the inequality is \( x < -1.25 \) rather than \( x \leq -1.25 \), you would use an open circle.

After marking '-1.25' with an open circle, you then shade the number line to the left to show that every number less than '-1.25' is a solution to the inequality. This visual guide helps students quickly see which numbers satisfy the inequality and provides a clear method to check solutions.

Number Line Representation

A number line is a powerful tool for representing inequalities, giving a clear and immediate visual cue to the range of solutions. When you have determined the solution to an inequality, such as \( x < -1.25 \), representing it on a number line confirms your results.

In the step-by-step solution, after isolating the variable and finding that \( x < -1.25 \), the next task is to translate this information onto a number line. Here is how to do that:

• Draw a horizontal line with arrows on both ends to signify continuation in both the positive and negative directions.
• Position numbers on the line which include the value that represents the boundary of your inequality (i.e., '-1.25').
• Choose an open or closed circle to mark this boundary based on whether the inequality includes the value as a solution.
• Shade in the side that represents the solution to the inequality, indicating that every point on the shaded side is a part of the solution set.

In our example, the open circle on '-1.25' and the shading to the left demonstrate that all numbers to the left of '-1.25' on the number line are solutions to the inequality. Such a representation is not just aligned with mathematical conventions but also supports a student's comprehension of the infinite set of solutions that inequalities often possess.