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Chapter 3: Problem 26

Find the values of \(x\) that solve the inequality. $$|1-4 x|>7$$

Short Answer

Expert verified
The solution is \(x < -\frac{3}{2}\) or \(x > 2\).

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \(|1-4x| > 7\) means the absolute value of \(1-4x\) is greater than 7. Recall that for an absolute value inequality like \(|A| > B\), the solution is found from two inequalities: \(A > B\) or \(A < -B\).
02

Set Up the First Inequality

Set up the first inequality by removing the absolute value: \(1 - 4x > 7\).
03

Solve the First Inequality

Start by solving \(1 - 4x > 7\):Subtract 1 from each side:\[-4x > 6\]Divide both sides by \(-4\), remembering to reverse the inequality sign:\[x < -\frac{3}{2}\]
04

Set Up the Second Inequality

Set up the second inequality from the absolute value: \(1 - 4x < -7\).
05

Solve the Second Inequality

Solve \(1 - 4x < -7\):Subtract 1 from each side:\[-4x < -8\]Divide both sides by \(-4\), reversing the inequality sign:\[x > 2\]
06

Combine the Solutions

The solutions to the inequalities \(1 - 4x > 7\) and \(1 - 4x < -7\) combine to give two separate ranges, which do not overlap. Thus, the solution to the original inequality \(|1 - 4x| > 7\) is \(x < -\frac{3}{2}\) or \(x > 2\).

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is a measure of magnitude without regard to sign. For any real number \(A\), the absolute value is denoted by \(|A|\) and is defined as:
  • \(|A| = A\) if \(A \geq 0\)
  • \(|A| = -A\) if \(A < 0\)
This property is crucial when it comes to solving equations and inequalities. The function \(|1-4x|>7\) involves determining the values of \(x\) for which the expression \(1-4x\) has a magnitude greater than 7. It essentially translates into two cases where the expression can be either greater than 7 or less than -7 due to the nature of absolute value.
Inequality Solution
Solving inequalities requires careful manipulation of expressions, often similar to solving equations but with special attention to the direction of the inequality sign. For the inequality \(|1-4x| > 7\), we consider two separate linear inequalities:
  • First inequality: \(1-4x > 7\)
  • Second inequality: \(1-4x < -7\)
These inequalities arise because the expression within the absolute value can lie either above 7 or below -7. Solving each inequality involves standard algebraic steps:
  • Subtract constants from both sides.
  • Divide or multiply coefficients of \(x\), being mindful to reverse the inequality sign when multiplying or dividing by a negative number.
This reversal reflects that multiplying or dividing by a negative swaps the order of numbers.
Inequalities in Mathematics
Inequalities are expressions that show the relative size or order of two values. Unlike equations, they describe a range of possible solutions. There are four primary inequality symbols:
  • Greater than: \(>\)
  • Less than: \(<\)
  • Greater than or equal to: \(\geq\)
  • Less than or equal to: \(\leq\)
In solving inequalities, the main goal is to isolate the variable on one side. Special care must be taken when multiplying or dividing both sides of an inequality by a negative number, as this action reverses the direction of the inequality sign. Inequalities often have more than one solution, representing a range of values rather than a single number.
Compound Inequalities
Compound inequalities link two or more inequalities together, often indicating a range of solutions. They can be described using the word "or" (disjunction) or "and" (conjunction). For an inequality of the form \(|1-4x|>7\), we separate it into two parts:
  • \(x < -\frac{3}{2}\) or
  • \(x > 2\)
This particular expression describes a situation where \(x\) can lie in either of the two separate regions on the number line. There is no overlap, so the solution isn't a continuous interval but two distinct sections. Compound inequalities facilitate a wider understanding of possible solutions, often leading to split intervals on the real number line.

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