Solve the following inequality: [tex]|x+6| \leq 4[/tex]
Answer (2)
We are given the absolute value inequality ∣ x + 6∣ ≤ 4 .
We rewrite the inequality as − 4 ≤ x + 6 ≤ 4 .
Subtracting 6 from all parts, we get − 10 ≤ x ≤ − 2 .
The solution to the inequality is − 10 ≤ x ≤ − 2 .
Explanation
Understanding the Problem We are given the inequality ∣ x + 6∣ ≤ 4 . Our goal is to find all values of x that satisfy this inequality.
Breaking Down the Absolute Value The absolute value inequality ∣ x + 6∣ ≤ 4 means that the distance between x + 6 and 0 is less than or equal to 4. This can be written as two separate inequalities:
− 4 ≤ x + 6 ≤ 4
Isolating x To solve for x , we subtract 6 from all parts of the inequality:
− 4 − 6 ≤ x + 6 − 6 ≤ 4 − 6
− 10 ≤ x ≤ − 2
The Solution The solution to the inequality is − 10 ≤ x ≤ − 2 . This means that x can be any value between -10 and -2, inclusive.
Examples
Absolute value inequalities are useful in many real-world situations. For example, if you are manufacturing parts for a machine, you might have a tolerance for the size of the parts. If the tolerance is ± 0.01 inches, then the actual size of the part must be within 0.01 inches of the specified size. This can be expressed as an absolute value inequality. Another example is in finance, where you might want to keep your investment within a certain range of values. Absolute value inequalities can help you determine the range of values that your investment can take.
The solution to the inequality ∣ x + 6∣ ≤ 4 is − 10 ≤ x ≤ − 2 , meaning that x can be any value between -10 and -2, inclusive. To solve, we first rewrite the inequality without the absolute value and then isolate x by subtracting 6. Thus, the final answer shows the range of values that satisfy the original condition.
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