Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, 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 \(( -2, \frac{16}{3} ]\).

Step 1: Understand the Compound Inequality

The given compound inequality is \(-4 < 3x + 2 \leq 18\). This implies that the expression \(3x + 2\) is greater than \(-4\) and at the same time less than or equal to \(18\). We need to solve two parts simultaneously: \(-4 < 3x + 2\) and \(3x + 2 \leq 18\).

Step 2: Solve the First Part of the Inequality

Consider the inequality \(-4 < 3x + 2\). Subtract 2 from both sides to isolate the term with \(x\):\[-4 - 2 < 3x\]\[-6 < 3x\]Divide both sides by 3 to solve for \(x\):\[\frac{-6}{3} < x\]\[-2 < x\]

Step 3: Solve the Second Part of the Inequality

Consider the second part of the inequality \(3x + 2 \leq 18\). Subtract 2 from both sides:\[3x + 2 - 2 \leq 18 - 2\]\[3x \leq 16\]Divide both sides by 3 to solve for \(x\):\[x \leq \frac{16}{3}\]

Step 4: Combine Solutions from Both Inequalities

Combine the results from Step 2 and Step 3. We have;\[-2 < x\]and \[x \leq \frac{16}{3}\]Thus, the compound inequality is satisfied for:\(-2 < x \leq \frac{16}{3}\)

Step 5: Express the Solution in Interval Notation

The solution obtained from combining the inequalities is \(-2 < x \leq \frac{16}{3}\). In interval notation, this is represented as \(( -2, \frac{16}{3} ]\). This tells us \(x\) is greater than \(-2\) and less than or equal to \(\frac{16}{3}\).

Key Concepts

Interval Notation

Interval notation is a way of expressing a set of numbers as an interval on the number line. It is often used to describe the solution sets for inequalities in a more compact form.
For example, in the inequality \(-2 < x \leq \frac{16}{3}\), we describe all the numbers that satisfy this condition using interval notation as\(( -2, \frac{16}{3} ]\).

• The parenthesis "(" next to the -2 means that -2 is not included in the set of solutions. This represents "greater than" in the inequality.
• The bracket "]" next to \(\frac{16}{3}\) means that \(\frac{16}{3}\) is included in the set of solutions, which signifies "less than or equal to".

Interval notation provides a clear, concise way to present the range of solutions. Make sure to pay attention to the symbols as they indicate whether endpoints are included or excluded.
This notation is very handy, especially when dealing with solutions of compound inequalities like in our exercise. It helps you quickly understand the range of numbers being described without needing to interpret multiple statements.

Algebraic Solutions

Algebraic solutions are the steps used to find values of variables that satisfy given conditions. In the context of inequalities, it involves manipulating the inequality to isolate the variable.
Let's break down the process using a simple example:

• Start with an inequality, for instance, \(-4 < 3x + 2\). Your goal is to isolate "\(x\)".
• Subtract 2 from both sides to get\(-6 < 3x\).
• Divide by 3 to completely isolate "\(x\)", resulting in\(-2 < x\).
• Repeat similar steps for any accompanying part of the compound inequality. For example, the inequality\(3x + 2 \leq 18\) becomes \(x \leq \frac{16}{3}\) after similar manipulations.

These steps ensure that "\(x\)" is left on one side of the inequality. By carefully performing operations on both sides, you accurately discover the range of feasible solutions.
Algebraic solutions form the backbone of solving inequalities, making it essential to grasp each operation and its role in transforming the inequality.

Inequality Solving Steps

Solving inequalities involves specific steps that are similar to solving equations but with extra attention to maintaining the inequality's direction.
The steps to solve compound inequalities like the one in our exercise are as follows:

• Break down the compound inequality into separate inequalities. Our example started as two parts: \(-4 < 3x + 2\) and \(3x + 2 \leq 18\).
• Solve each part separately:
  • For the first part, isolate "\(x\)" by subtracting and dividing, resulting in \(-2 < x\).
  • For the second part, perform similar steps to find \(x \leq \frac{16}{3}\).
• Combine the solutions by ensuring they satisfy both conditions simultaneously. In our case, this means finding "\(x\)" values for the combined inequality \(-2 < x \leq \frac{16}{3}\).
• Translate the combined results into interval notation for clarity and simplicity. Here, the solution is given as \(( -2, \frac{16}{3} ]\).

These steps are crucial as they help to understand the boundaries within which the solutions lie. Paying attention to small details like inequality symbol direction and combining properly will lead to the correct solution.Understanding and methodically following these steps will enable you to solve compound inequalities with confidence.