Problem 58 - 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. \(3 x+2 \leq 14\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x ≤ 4\), represented on the number line by a closed circle at 4 and a line extending towards the negative numbers.

Step 1: Isolate the term with x

The first step is to isolate the 3x term on one side of the inequality. This can be done by subtracting 2 from both sides of the inequality. Hence, the inequality becomes: \(3x ≤ 14 - 2\) which simplifies to \(3x ≤ 12\).

Step 2: Solve for x

Next, to find the solution to the inequality, divide both sides by 3, applying the division property of inequality. The resulting inequality is \(x ≤ 12/3\).

Step 3: Simplify

Simplify the value to get the final inequality as \(x ≤ 4\). This is the solution to the original inequality.

Step 4: Graphing on a number line

On a number line, the number 4 and all other numbers less than or equal to 4 would be the solution set. To graph this, one could place solid circle on 4 (indicating it's part of solution) and draw a line extending towards left (indicating all numbers less than 4 are part of the solution).

Key Concepts

Addition Property of Inequality

The addition property of inequality is a key concept that allows us to manipulate inequalities while maintaining their truth. When working with inequalities, it's important to remember that you can add or subtract the same value from both sides without changing the inequality's direction.

• If you have an inequality such as \(a < b\), adding the same number \(c\) to both sides will keep the inequality true: \(a + c < b + c\).

In our original exercise, to isolate the term with \(x\), we subtracted 2 from both sides of the inequality \(3x + 2 \leq 14\). By doing this, the equality adjusted to \(3x \leq 12\). Since we subtracted equally from both sides, the inequality stayed balanced and true. This property is foundational in making sure our manipulations do not alter the balance or direction of the inequality.

Multiplication Property of Inequality

Once you have simplified the inequality using addition or subtraction, the next step often involves the multiplication property of inequality to solve for the variable.
The multiplication (or division) property states that you can multiply or divide both sides of an inequality by a positive number without changing the direction of the inequality. However, if you multiply or divide by a negative number, you must reverse the inequality sign.
In our step-by-step solution, after getting \(3x \leq 12\), we divided both sides by 3, a positive number, to solve for \(x\), resulting in \(x \leq 4\). Since division by a positive number does not change the inequality's direction, the inequality remained \(\leq\). This property ensures the inequality's integrity while isolating the variable, a crucial step in solving inequalities.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of the solution set, helping one understand the range of values that satisfy the inequality.

• Begin by identifying the critical value(s) from the inequality; in our example, it is 4 from \(x \leq 4\).
• Mark this point on the number line. For \(\leq\) or \(\geq\) inequalities, use a solid or closed circle to indicate that the point is included in the solution set.

Since the inequality is \(x \leq 4\), every value of \(x\) that is 4 or less meets the inequality's requirements. Accordingly, the number line will display a solid circle over the number 4, with a shaded line extending to the left, illustrating all numbers less than 4.

Number Line Representation

The number line representation is an effective tool to visually convey the solutions of inequalities, making it easier to comprehend the scope of possible answers.

• Start at the critical value you identified. Here, it is 4.
• Decide on the type of circle. Use a filled circle at 4 because \(x \leq 4\) includes 4 as a valid solution.
• Draw a line extending to the left, covering all numbers less than 4, to represent all possible solutions.

This step transforms the abstract solution of \(x \leq 4\) into a visual form. It shows a solid span from negative infinity up to, and including, 4. Visual aids like this can bridge understanding by visually demonstrating how inequalities encompass ranges of numbers.