In Exercises 59–94, solve each absolute value inequality. |x + 3| ≤ 4

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1. Recall that the absolute value inequality \(|A| \leq B\) means that the expression inside the absolute value, \(A\), lies between \(-B\) and \(B\). So, rewrite the inequality \(|x + 3| \leq 4\) as a compound inequality: \(-4 \leq x + 3 \leq 4\).
2. Next, solve the compound inequality by isolating \(x\). Subtract 3 from all three parts of the inequality: \(-4 - 3 \leq x + 3 - 3 \leq 4 - 3\), which simplifies to \(-7 \leq x \leq 1\).
3. Interpret the solution: \(x\) is any number between \(-7\) and \$1$, inclusive, because the inequality is 'less than or equal to'.
4. Express the solution in interval notation as \([-7, 1]\), which represents all \(x\) values from \(-7\) to \$1$ including the endpoints.
5. Optionally, you can graph the solution on a number line by shading the region between \(-7\) and \$1$ and including solid dots at these points to indicate they are part of the solution.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative. Understanding this helps in interpreting and solving absolute value inequalities.

Solving Absolute Value Inequalities

An inequality involving absolute value, such as |A| ≤ B, can be rewritten as a compound inequality: -B ≤ A ≤ B. This approach transforms the absolute value inequality into two linear inequalities that can be solved simultaneously to find the solution set.

Compound Inequalities

Compound inequalities involve two inequalities joined by 'and' or 'or'. For absolute value inequalities like |x + 3| ≤ 4, the solution requires finding all x values that satisfy both -4 ≤ x + 3 and x + 3 ≤ 4 simultaneously, resulting in a range of solutions.