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Chapter 2: Problem 56

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -4<3-x<4 $$

Short Answer

Expert verified
The solution is \((-1, 7)\).

Step by step solution

01

Break Down the Compound Inequality

The compound inequality \(-4 < 3 - x < 4\) can be split into two separate inequalities: 1. \(-4 < 3 - x\) 2. \(3 - x < 4\). We'll solve each inequality step by step.
02

Solve the First Inequality

Start with the inequality \(-4 < 3 - x\). First, subtract 3 from both sides: \(-4 - 3 < -x\) which simplifies to:\(-7 < -x\). Next, multiply each side by -1 to solve for \(x\), remembering to reverse the inequality sign:\(7 > x\) or equivalently, \(x < 7\).
03

Solve the Second Inequality

Now solve the inequality \(3 - x < 4\). Subtract 3 from both sides:\(-x < 1\).Multiply each side by -1 and reverse the inequality:\(x > -1\).
04

Combine the Solutions

With \(x > -1\) from Step 3 and \(x < 7\) from Step 2, you can combine these two results to form the solution of the compound inequality.This gives:\(-1 < x < 7\).
05

Express the Solution in Interval Notation

Translate the compound inequality \(-1 < x < 7\) into interval notation. The interval notation is: \((-1, 7)\). This represents all numbers between -1 and 7, not including the endpoints.

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Key Concepts

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

Interval Notation
Interval notation is a way of expressing subsets of real numbers on the number line. It provides a concise way to describe the range of solutions for inequalities. By using brackets and parentheses, we indicate whether endpoints are included or excluded from the solution set. For example, if a solution includes all numbers between -1 and 7, excluding -1 and 7 themselves, it's written as \((-1, 7)\).
  • Parentheses \(()\) are used when the endpoints are not included, meaning the values are "open."
  • Brackets \([]\) are used to include the endpoints, indicating "closed" values.
Interval notation is beneficial for solving, interpreting, and graphing inequalities on a number line. It makes it easier to visualize the scope of numbers that satisfy a particular inequality.
Solving Inequalities
Solving inequalities is similar to solving equations, with a few additional steps. To find the solution for an inequality like \(3 - x < 4\), we perform operations common to solving equations:
  • Adding or subtracting numbers from both sides.
  • Multiplying or dividing both sides by a positive number.
However, one crucial rule when solving inequalities involves reversing the inequality sign when multiplying or dividing by a negative number.
For example:
  • If you have \(-x < 1\), multiplying by -1 to make \(x\) positive requires flipping the < symbol to >, resulting in \(x > -1\).
Always ensure to perform the same operation on both sides of the inequality. This maintains the balance and truth of the inequality.
Reversing Inequality Signs
Reversing inequality signs is a vital concept in solving inequalities. This normally occurs when you multiply or divide each side of an inequality by a negative number. It ensures the inequality relation remains accurate. For example, when dealing with \(-x < 1\), multiplying both sides by -1 yields \(x > -1\). The inequality sign is reversed from < to > during this operation.
  • Make sure to reverse the inequality sign any time you multiply or divide by a negative value.
  • Remember that this step is crucial to keeping the inequality true.
Failing to reverse the sign when needed can lead to incorrect solutions, so always double-check this step when solving inequalities.

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Most popular questions from this chapter

Solve each equation. $$ |x+1|=|x-4| $$

Solve each inequality. $$ \left|\frac{2 x+1}{2}\right|>1 $$

For Problems \(55-64\), solve each equation and inequality by inspection. $$ |3 x-1|>-2 $$

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x| \leq 2 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -6<4 x-5<6 $$
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