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Chapter 0: Problem 76

Solve each absolute value inequality. $$\left|3-\frac{3}{4} x\right|>9$$

Short Answer

Expert verified
The solution to the inequality is \(x > 16\) or \(x < -8\).

Step by step solution

01

Set Up Two Inequalities

The given absolute value inequality is \(|3 - \frac{3}{4}x| > 9\). Using the rule of absolute value, this can be broken down into two separate inequalities: \(3 - \frac{3}{4}x > 9\) and \(3 - \frac{3}{4}x < -9\).
02

Solve the First Inequality

First, solve \(3 - \frac{3}{4}x > 9\), isolating x. Subtract 3 from both sides which gives \(-\frac{3}{4}x > 6\). Then, divide both sides by \(-\frac{3}{4}\), which gives \(x < -8\). Please remember, the inequality sign changes direction when both sides are divided by a negative number.
03

Solve the Second Inequality

Next, solve \(3 - \frac{3}{4}x < -9\), isolating x. Subtract 3 from both sides which gives \(-\frac{3}{4}x < -12\). Then, divide both sides by \(-\frac{3}{4}\), which gives \(x > 16\). Again, note that the inequality sign changes direction due to the division by a negative number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Inequalities
Inequalities are an essential concept in algebra, expressing a range of possible solutions rather than a single solution. An inequality compares two values, determining if one is greater, less than, or possibly equal to another. Inequalities use symbols like:
  • ">" (greater than)
  • "<" (less than)
  • ">=" (greater than or equal to)
  • "<=" (less than or equal to)

In an absolute value inequality, the absolute value function influences the inequality by determining distance from zero, without considering direction. By understanding inequalities, you appreciate solving for a variable's range rather than a specific value. This foundational knowledge helps prepare for more complex algebra problems.
Decoding Algebraic Expressions
Algebraic expressions are crucial for translating real-world problems into mathematical language. They consist of numbers, variables, and operation symbols, representing quantity and relationships. The expression \(3 - \frac{3}{4}x\) is an excellent example of how these elements combine.
  • "3" is a constant, a fixed value.
  • "\(\frac{3}{4}\)x" is a term involving a variable ("\(x\)") and a coefficient ("\(\frac{3}{4}\)").

When working with algebraic expressions in inequalities, it becomes essential to see how these components affect the expression. Isolating the variable often helps to find solutions or to better understand the relationships represented by the expression.
Mastering Solving Equations
Solving equations is about finding values for variables that make the equation true. The process involves manipulating the equation using operations like addition, subtraction, multiplication, and division.
Consider the inequality \(3 - \frac{3}{4}x > 9\). Solving this involves several steps:
  • Subtract 3 from both sides to isolate terms with the variable.
  • Divide by \(-\frac{3}{4}\) to find \(x\). Remember to flip the inequality when dividing by a negative number.

Similarly, for \(3 - \frac{3}{4}x < -9\), the same operations apply, ensuring careful attention to changing the inequality direction. Mastering equation solving includes comprehending these operations and their impacts, leading to accurate and reliable results.

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Most popular questions from this chapter

List all numbers that must be excluded from the domain of each rational expression. $$\frac{7}{2 x^{2}-8 x+5}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

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