/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 Solve and graph. $$ |x-3|+2&... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 71

Solve and graph. $$ |x-3|+2>7 $$

Short Answer

Expert verified
The solution is \(x < -2\) or \(x > 8\).

Step by step solution

01

- Isolate the Absolute Value

Start by isolating the absolute value expression. Subtract 2 from both sides of the inequality: \(|x-3| + 2 - 2 > 7 - 2\). This simplifies to \(|x-3| > 5\).
02

- Set Up Inequality Cases

The absolute value inequality \(|x-3| > 5\) means there are two cases to consider: 1. \(x - 3 > 5\) 2. \(x - 3 < -5\)
03

- Solve Each Case

Solve each inequality separately: 1. For \(x - 3 > 5\), add 3 to both sides: \(x > 8\).2. For \(x - 3 < -5\), add 3 to both sides: \(x < -2\).
04

- Combine Solutions

The solution to the inequality is the union of both cases. Therefore, the solution is \(x < -2\) or \(x > 8\).
05

- Graph the Solution

On a number line, draw two open circles at \(-2\) and \(8\) to denote that these points are not included. Shade the regions to the left of \(-2\) and to the right of \(8\) to indicate all the values that satisfy the inequality.

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Key Concepts

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

inequality solving
Solving inequalities involves finding the values of the variable that make the inequality true. For the inequality \(|x-3|+2>7\), the goal is to isolate the absolute value expression first. Subtract 2 from both sides of the inequality: \(|x-3| > 5\). This separation helps us identify two separate inequality cases.
graphing inequalities
Graphing inequalities visually represents the solution on a number line. For the inequality \(|x-3| > 5\), the solutions are two separate regions: \(x < -2\) and \(x > 8\). \To graph this, use open circles at \(-2\) and \(\textbf{8}\) to show that these points are not included. \Shade the regions to the left of \(-2\) and to the right of \(\textbf{8}\) to illustrate the solution clearly. This visual helps students quickly identify all possible values satisfying the inequality.
absolute value
Absolute value measures the distance of a number from zero, regardless of direction. \For \(|x-3| > 5\), it means the distance between \(x\) and 3 is greater than 5 units. We set up two cases: \
    \
  • \((x - 3) > 5\): Here, \((x\) should be more than 8.
    • \
  • \(x - 3 < -5\): Here, \(x\) should be less than -2.
    • \
\This dual case setup is crucial because absolute values account for distances in both positive and negative directions.

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Most popular questions from this chapter

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