Problem 40 - Algebra for College Students

Question

Solve each equation and inequality. \(|-3 x-4| \leq 15\)

Step-by-Step Solution

Answer

The solution is \(-\frac{19}{3} \leq x \leq \frac{11}{3}\).

Step 1: Understanding the Absolute Value Inequality

The inequality \(|-3x-4| \leq 15\) is an absolute value inequality, which means the expression inside the absolute value can be between a range of values. Thus, this inequality states that the expression \(-3x-4\) should be between \-15 and 15. We can split this into two linear inequalities:1. \(-3x-4 \leq 15\)2. \(-3x-4 \geq -15\)

Step 2: Solve the First Linear Inequality

Let's solve the first inequality \(-3x-4 \leq 15\):1. Add 4 to both sides: \(-3x \leq 19\)2. Divide by -3, and remember to reverse the inequality sign: \(x \geq -\frac{19}{3}\)

Step 3: Solve the Second Linear Inequality

Now, let's solve the second inequality \(-3x-4 \geq -15\):1. Add 4 to both sides: \(-3x \geq -11\)2. Divide by -3, and reverse the inequality sign: \(x \leq \frac{11}{3}\)

Step 4: Write the Solution as a Compound Inequality

Since both conditions must be true, combine them into a compound inequality:\[-\frac{19}{3} \leq x \leq \frac{11}{3}\]

Step 5: Interpretation of the Result

The solution \(-\frac{19}{3} \leq x \leq \frac{11}{3}\) indicates that any value of \(x\) within this range will satisfy the original inequality \(|-3x-4| \leq 15\).

Key Concepts

Linear Inequalities

Linear inequalities are expressions where one side is not equal to the other. They use inequality symbols such as \(<, \leq, >\), and \(\geq\), to show the relationship between the expressions. For example, the inequality \(-3x - 4 \leq 15\) indicates that the left-hand side is less than or equal to the right-hand side.

When solving linear inequalities, similar to linear equations, follow these steps:

• Perform the same operations on both sides, like addition, subtraction, multiplication, or division.
• Remember to change the direction of the inequality when multiplying or dividing by a negative number.

These steps ensure the inequality remains true. With our example, solving \(-3x - 4 \leq 15\) involved adding 4 to both sides and then dividing by \(-3\), which reversed the inequality to \(x \geq -\frac{19}{3}\).

Compound Inequalities

Compound inequalities are two or more inequalities combined using the words "and" or "or." They allow us to represent a range of values that satisfy two conditions simultaneously. In our exercise, we combined the two linear inequalities: \(-3x - 4 \leq 15\) and \(-3x - 4 \geq -15\), into one compound inequality.

The compound inequality \(-\frac{19}{3} \leq x \leq \frac{11}{3}\) shows all possible solutions that satisfy both conditions at the same time. This means that any \(x\) value within the range \(-\frac{19}{3}\) to \(\frac{11}{3}\) will make the original absolute value inequality true. Compound inequalities help provide a clearer understanding of where the solutions lie on a number line.

Inequality Solving Steps

The steps for solving inequalities are crucial for finding the range of solutions that satisfy the given conditions. Let's break down these steps using our example inequality \(|-3x - 4| \leq 15\):

• Understand the Inequality: Recognize the type of inequality – here, it's an absolute value inequality meaning \(-3x - 4\) lies between two values: -15 and 15.
• Split into Linear Inequalities: Turn the absolute value inequality into two separate linear inequalities: \(-3x - 4 \leq 15\) and \(-3x - 4 \geq -15\).
• Solve Each Inequality: Solve each linear inequality independently to find the valid range of values for \(x\).
• Combine the Solutions: Once both inequalities are solved, combine the solutions to form a compound inequality which identifies the values that satisfy both conditions.
• Interpret the Result: Clearly state what the solution range means in context – it describes all \(x\) values that make the original inequality true.

These systematic steps ensure a thorough and accurate solution, making the task of solving inequalities manageable and clear.