Problem 78 - 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. \(\frac{x}{4}-3 \geq 1\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 16\). This means that any number 16 or greater will satisfy the inequality.

Step 1: Isolate the term with the variable

The first step is to isolate the term \(\frac{x}{4}\) by adding 3 to both sides of the inequality. This results in: \(\frac{x}{4} \geq 1 + 3\). Simplifying gives us: \(\frac{x}{4} \geq 4\).

Step 2: Solve for the variable

Continue by isolating the variable \(x\), which requires getting rid of the fraction. To do this we multiply both sides of the inequality by 4. This gives us: \(x \geq 4*4\). Simplifying, we find: \(x \geq 16\).

Step 3: Graph the solution set on a number line

Now we need to represent the solution \(x \geq 16\) on a number line. A circle is drawn on 16 and, as it's greater than or equal to, it is filled in. Then, because \(x\) is greater than or equal to 16, all points to the right of 16 on the number line are also included. This shows the solution set for the inequality is \(x \geq 16\).

Key Concepts

Understanding the Addition Property of Inequality

The addition property of inequality is a fundamental concept used to solve inequalities. It states that you can add the same number to both sides of an inequality without changing the inequality's direction. This is similar to a balance where you are adding equal weights on both sides.
For example, if you have an inequality like \(a \leq b\), adding a number \(c\) to both sides gives you \(a + c \leq b + c\). This property is crucial when you're trying to isolate a variable, as seen in our original exercise.
In the exercise \(\frac{x}{4} - 3 \geq 1\):

• We use the addition property to eliminate the \(-3\) by adding \(3\) to both sides of the inequality.
• This helps in isolating the variable term \(\frac{x}{4}\).

By performing this addition, the inequality becomes \(\frac{x}{4} \geq 4\), simplifying our task of solving for \(x\). Remember, always make sure the terms you add or subtract help in isolating the variable effectively.

Exploring the Multiplication Property of Inequality

After using the addition property to simplify an inequality, you may need the multiplication property to solve for the variable completely. The multiplication property of inequality states that if you multiply or divide an inequality by a positive number, the direction of the inequality does not change.
For instance, with \(a \geq b\), multiplying both sides by a positive value \(d\) results in \(ad \geq bd\). However, remember, if you multiply or divide by a negative number, you must flip the inequality sign.
In the problem we worked on, we used this property:

• We multiplied both sides of \(\frac{x}{4} \geq 4\) by 4 to get rid of the fraction and directly solve for \(x\).
• This led to \(x \geq 16\), preserving the inequality's direction since 4 is positive.

Pay attention to the sign of the number you multiply when solving inequalities, as it determines whether the inequality sign should flip.

Graphing Inequalities on a Number Line

Graphing inequalities is a visual way to represent solution sets. It provides a clear picture of which numbers are included in the solution.
To graph an inequality like \(x \geq 16\), do the following:

• Draw a number line that includes 16.
• Place a solid filled-in circle at 16, indicating that 16 is part of the solution (due to the "or equal to" part of the inequality).
• Draw an arrow from 16 going to the right, symbolizing all numbers greater than 16 are solutions.

This graphical representation helps in quickly interpreting the range of possible values for \(x\). It makes it easy to see that any number 16 and onward satisfies the inequality \(x \geq 16\). Remember not to fill the circle if it were a strict inequality (\(>\) or \(<\)), as the endpoint wouldn't be included in the solutions.