The temperatures on the surface of Mars in degrees Celsius approximately satisfy the inequality ∣C+84∣≤6 |C + 84 | ≤ 6. What range of temperatures corresponds to this inequality?

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 78

Chapter 2, Problem 78

The temperatures on the surface of Mars in degrees Celsius approximately satisfy the inequality $∣ C + 84 ∣ \leq 6$ . What range of temperatures corresponds to this inequality?

Verified step by step guidance

Recognize that the inequality involves an absolute value: $|C + 84| \leq 6$. This means the expression inside the absolute value, $C + 84$, is at most 6 units away from 0 on the number line.

Rewrite the absolute value inequality $|C + 84| \leq 6$ as a compound inequality without the absolute value: $-6 \leq C + 84 \leq 6$.

Isolate the variable $C$ by subtracting 84 from all parts of the inequality: $-6 - 84 \leq C + 84 - 84 \leq 6 - 84$.

Simplify the inequality to get the range for $C$: $-90 \leq C \leq -78$.

Interpret the result: the temperatures $C$ on the surface of Mars lie between $-90$ degrees Celsius and $-78$ degrees Celsius, inclusive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequality

An absolute value inequality like |x| ≤ a means the distance of x from zero is at most a. It can be rewritten as a compound inequality: -a ≤ x ≤ a. This helps in finding the range of values that satisfy the inequality.

Translating Absolute Value Inequalities

When the absolute value involves an expression, such as |C + 84| ≤ 6, rewrite it as -6 ≤ C + 84 ≤ 6. This isolates the variable by breaking the inequality into two parts, making it easier to solve for C.

Solving Compound Inequalities

To solve -6 ≤ C + 84 ≤ 6, subtract 84 from all parts to isolate C, resulting in -90 ≤ C ≤ -78. This gives the range of temperatures that satisfy the original inequality.

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