Problem 55 - Introductory Algebra for College Students

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. \(-x<4\)

Step-by-Step Solution

Answer

The solution of the inequality \(-x<4\) is \(x<4\). On a number line, this is represented by an open circle at 4, with an arrow pointing to the left indicating all numbers less than 4.

Step 1: Rewrite Inequality

Rewrite the inequality as \(-x> -4\). Here, multiplication property of inequalities is used, where multiplying or dividing an inequality by a negative number flips the inequality sign.

Step 2: Simplify Inequality

The inequality \(-x> -4\) can be simplified by dividing both sides by \(-1\). This yields \(x<4\).

Step 3: Graph the Solution

Draw a number line. Draw an open circle at 4 to illustrate that 4 is not included in the solution set. Draw an arrow from 4 to the left to indicate all numbers less than 4 form the solution set.

Key Concepts

Inequality Solution

Inequality solutions are all about finding the set of numbers satisfying the given inequality condition. In this exercise, we start with the inequality \(-x < 4\). To solve it, we need to isolate the variable \(x\). Here, the multiplication property of inequality comes into play.

• This property states that when you multiply or divide both sides of an inequality by the same negative number, the direction of the inequality sign must be flipped.
• In our exercise, we multiply both sides by \(-1\) which reverses the inequality from \(-x < 4\) to \(x > -4\).

Breaking each problem down in this manner helps in understanding the underlying principles of solving inequalities with negative coefficients. By switching signs appropriately, we can find the correct range of values for our variable. Such logical steps ensure that we respect the mathematical integrity of inequalities.

Number Line Graph

A number line graph visually represents solutions to inequalities. This visual tool can clarify more abstract numerical concepts. In our problem, once the inequality is solved to \(x > -4\), it's time to use the number line to illustrate this.

• First, plot a point at the critical number from your inequality, which is \(-4\) in this case. Instead of a solid dot, use an open circle to show that \(-4\) is not included in the solution.
• Then, draw an arrow pointing to the right of this number on the line. This arrow indicates all the values greater than \(-4\), aligning perfectly with the algebraic solution \(x < 4\).

The number line gives you a complete picture of the solution set, demonstrating the range of possible solutions in a clear, concise manner. Always double-check to ensure the correct direction and method of indication are used, such as open or closed circles and arrow directions.

Negative Coefficient Inequality

Working with negative coefficient inequalities poses a common challenge due to sign reversal. The key principle to remember is how the inequality sign flips its direction when you multiply or divide both sides by a negative number. This inherent flip is due to the properties of numbers on the number line.

• Start with the inequality, acknowledging the negative coefficient on \(x\), such as \(-x < 4\).
• When removing the negative by dividing by \(-1\), remember to flip the inequality sign: the result becomes \(x > -4\).

This practice is vital as it affects the accuracy of the solution. Ignoring the rule of changing the inequality direction leads to incorrect solutions. Practice this careful check whenever dealing with such inequalities. Understanding this concept thoroughly prepares you to tackle more complex algebraic problems with confidence.