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Chapter 1: Problem 62

In Exercises 59鈥�94, solve each absolute value inequality. $$ |x+3| \leq 4 $$

Short Answer

Expert verified
The solution to the absolute value inequality \(|x+3| \leq 4\) is \(-7 \leq x \leq 1\)

Step by step solution

01

Interpret the Absolute Value as a Distance

We can interpret the absolute value as a distance. The equation \(|x+3| \leq 4\) means the distance between \(x\) and \(-3\) on the number line is less than or equal to 4.
02

Solve for the Inequality

We then create two equations from the inequality. One for the positive distance i.e. \(x+3 \leq 4\) and one for the negative distance i.e. \(-(x+3) \leq 4\). Solving the first equation \(x+3 \leq 4\) gives \(x \leq 1\) and the second equation \(-(x+3) \leq 4\) gives \(-x -3 \leq 4\) or \(x \geq -7\).
03

Combine the solutions

The solution to the inequality is the intersection of the solutions for the two equations. The solutions of the inequalities from Step 2 are \(-7 \leq x \leq 1\)

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Key Concepts

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

Distance Interpretation
Understanding an absolute value inequality begins with interpreting it as a distance. The absolute value expression \(|x+3|\) can be seen as measuring how far the number \(x\) is from \(-3\) on a number line. In our inequality, \(|x+3| \leq 4\), the symbol \(\leq 4\) indicates that this distance is at most 4 units. This visual interpretation helps you think of absolute values not just as numbers but as positions on a line, allowing you to visualize the solutions needed for the inequality.
Inequality Solutions
Solving absolute value inequalities involves breaking down the problem into manageable parts. The inequality \(|x+3| \leq 4\) can be split into two separate inequalities: one representing the positive distance \(x+3 \leq 4\) and the other, the negative distance \(-(x+3) \leq 4\). By solving \(x+3 \leq 4\), you find that \(x \leq 1\). And by solving \(-x-3 \leq 4\), you find that \(x \geq -7\). These solutions correspond to both parts of the distance that \(x\) can be from \(-3\).
Number Line
Visualizing solutions on a number line is crucial when working with inequalities. After solving the separate inequalities for the absolute value \(|x+3| \leq 4\), place \(-7\) and \(1\) as points on the number line. The region between these points, \(-7\) to \(1\), represents all possible \(x\) values that satisfy the inequality. This helps ensure that you don't miss any solution possibilities. Things become more intuitive and apparent when you can see them represented visually. Remember, the solution is not just individual points but a continuous section on the line.
Compound Inequalities
An absolute value inequality often results in a compound inequality, which combines two inequalities. For the expression \(|x+3|\leq 4\), after evaluating the distances, we combine them into the compound inequality \(-7 \leq x \leq 1\). This effectively narrows down the values of \(x\) into a single, bounded range. The term 'compound' signifies that two criteria must be satisfied simultaneously, which, in this case, form a range of solutions rather than a single point. This range provides a concise form of representing all valid solutions for \(x\).

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