Question

Solve the inequality \(|3 x+2| \leq 4\)

Short answer

Answer: The inequality holds true for \(x\) in the interval \([-2, \frac{2}{3}]\).

Step-by-step solution

Break the absolute value inequality into two separate inequalities

When \((3x+2) \geq 0\), the inequality is \(3x + 2 \leq 4\). When \((3x+2) < 0\), the inequality is \(-(3x + 2) \leq 4\).

Solve the first inequality

To solve the inequality \(3x + 2 \leq 4\), we'll first isolate the x term by subtracting 2 from both sides: \(3x \leq 2\). Next, we'll divide both sides by 3: \(x \leq \frac{2}{3}\).

Solve the second inequality

To solve the inequality \(-(3x + 2) \leq 4\), we first multiply both sides by -1 and reverse the inequality symbol: \((3x + 2) \geq -4\). Then we isolate the x term by subtracting 2 from both sides: \(3x \geq -6\). Finally, we'll divide both sides by 3: \(x \geq -2\).

Combine solutions and write in interval notation

Since \(x \leq \frac{2}{3}\) and \(x \geq -2\), we can write the solution in interval notation as \([-2, \frac{2}{3}]\). This represents the values of \(x\) that satisfy the inequality \(|3x + 2| \leq 4\).

Key concepts

Inequality Solving

Solving inequalities is a lot like solving equations, but with a few extra rules. Inequalities tell us about the range of possible values while equations give us a specific value. Here, we're dealing with an **absolute value inequality**, which can be a bit tricky as it involves two different conditions wrapped in one expression. Absolute value expressions like \(|3x+2|\) can make an inequality have two parts, because absolute value measures distance, and distance is always positive.

To solve an absolute value inequality like \(|3x+2| \leq 4\), we break it down into two separate inequalities:

• One for when the expression inside is non-negative: \(3x + 2 \leq 4\).
• And one for when it is negative, which we flip by multiplying by -1, reversing the inequality sign: \(- (3x + 2) \leq 4\), which simplifies to \(3x + 2 \geq -4\).

Understanding these steps helps in breaking down the problem into manageable parts.

Interval Notation

After solving both inequalities, we need a simple way to express our solution set. That's where **interval notation** comes in handy. Interval notation provides a compact form of writing down all the answers within certain bounds, without listing them all individually.

For the inequality \(|3x+2| \leq 4\), we have found:

• Solutions from \(-2\) to \(rac{2}{3}\).

We write this span of numbers as \([-2, \frac{2}{3}]\). The brackets mean those endpoints, \(-2\) and \(rac{2}{3}\), are included in the solution (the equality part of "less than or equal to" allows this).

Interval notation is not only compact but also visually clear, showing the start and end of possibilities for \(x\).

Mathematical Problem Solving

Tackling mathematical problems is like solving a puzzle. There are steps and techniques to follow, but creativity and understanding how to connect parts are crucial. This exercise demonstrates a systematic approach toward **mathematical problem solving**:
1. **Understanding the Problem:** We identified the type of inequality we deal with: an absolute value. This prepared us for the two-part solution process.2. **Breaking Down the Problem:** We split the inequality \(|3x + 2| \leq 4\) into two simpler inequalities, making the problem much easier to handle.3. **Solving by Steps:** We used algebraic tools like subtraction and division to isolate \(x\) in both simpler inequalities.4. **Combining Results:** After solving, we combined our results to highlight the complete answer set using interval notation.By following these steps, we gain confidence in dealing with more complex mathematical problems in the future. Such structured problem solving not only builds mathematical skills but also enhances logical thinking.