Problem 28 - College Algebra

Question

Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. $$ -4<3 x+2 \leq 18 $$

Step-by-Step Solution

Answer

The solution is \(-2 < x \leq \frac{16}{3}\) or \((-2, \frac{16}{3}]\).

Step 1: Isolate the Variable in the First Inequality

The given compound inequality is \(-4 < 3x + 2 \leq 18\). Start by separating the two inequalities: \(-4 < 3x + 2\) and \(3x + 2 \leq 18\). For the first inequality, subtract 2 from both sides to isolate the term with \(x\): \(-4 - 2 < 3x\) which simplifies to \(-6 < 3x\).

Step 2: Solve for x in the First Inequality

Continue with \(-6 < 3x\). Divide both sides by 3 to solve for \(x\): \(-6/3 < x\), which simplifies to \(-2 < x\).

Step 3: Isolate the Variable in the Second Inequality

Now, focus on the second inequality \(3x + 2 \leq 18\). Subtract 2 from both sides to get: \(3x \leq 18 - 2\), which simplifies to \(3x \leq 16\).

Step 4: Solve for x in the Second Inequality

From \(3x \leq 16\), divide both sides by 3: \(x \leq 16/3\). This simplifies to \(x \leq \frac{16}{3}\).

Step 5: Combine Both Results

Combine the results from the two inequalities: \(-2 < x \leq \frac{16}{3}\). This forms the solution to the compound inequality.

Step 6: Express in Interval Notation

Convert the combined inequality \(-2 < x \leq \frac{16}{3}\) into interval notation. Since \(x\) is greater than \(-2\) and up to and including \(\frac{16}{3}\), the interval notation will be \((-2, \frac{16}{3}]\).

Key Concepts

Interval Notation

When expressing solutions to inequalities, interval notation is a concise method for displaying the range of possible values. This can be especially helpful for compound inequalities, where multiple conditions must be satisfied simultaneously. For our compound inequality

• -2 < x ≤ \(\frac{16}{3}\)
• In this inequality, \(x\) is any number between -2 (not included) and \(\frac{16}{3}\) (included).

Interval notation represents this as \((-2, \frac{16}{3}]\). Notice the use of parentheses and brackets:

• Parentheses \(()\) mean that the endpoint is not included in the solution set.
• Brackets \([]\) indicate the endpoint is included.

This notation helps clearly communicate both the boundaries and whether each boundary is part of the solution. Interval notation is a valuable tool in mathematics because it allows you to express complex ideas in a simple way, making it easier to read and understand.

Isolating Variables

Isolating variables is a fundamental step in solving inequalities or equations. The goal is to get the variable on one side of the inequality sign by itself. This is critical for solving and understanding the solution set.

• First, identify which terms include your variable and which do not.
• Use basic arithmetic operations like addition, subtraction, multiplication, or division to move terms lacking the variable to the other side.

For our compound inequality, we started with two conditions:

• For -4 < 3x + 2: subtract 2 from both sides.
• For 3x + 2 ≤18: again, subtract 2 from both sides.

After adjusting both inequalities, we end up with 3x standing alone on one side. Then, simplify by performing any necessary arithmetic operations, like division, to finally isolate \(x\). Isolating variables step-by-step ensures accurate solutions and clarity in solving both simple and compound inequalities.

Inequality Signs

In the world of inequalities, understanding inequality signs is crucial. They tell us how values compare to each other, and this understanding is pivotal when writing math solutions. Inequality signs include:

• \(<\): less than.
• \(≤\): less than or equal to.
• \(>\): greater than.
• \(≥\): greater than or equal to.

In a compound inequality like -4 < 3x + 2 ≤18, we encounter both < and ≤ signs. They tell us:

• Values are being compared from less than on one side, to less than or equal to, on the other side.
• Solving involves maintaining these inequalities when isolating the variable, to express comparisons accurately.

Misunderstanding these signs can lead to incorrect interpretations. It is important to perform the same operations on all parts of the inequality without reversing the inequality sign, unless multiplying or dividing by a negative number. Understanding these symbols helps in drafting solutions that are mathematically valid and reliable. This ensures you maintain the true relationships expressed in your inequalities.