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

Verified step by step guidance

1. 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.
2. Rewrite the absolute value inequality \(|C + 84| \leq 6\) as a compound inequality without the absolute value: \(-6 \leq C + 84 \leq 6\).
3. Isolate the variable \(C\) by subtracting 84 from all parts of the inequality: \(-6 - 84 \leq C + 84 - 84 \leq 6 - 84\).
4. Simplify the inequality to get the range for \(C\): \(-90 \leq C \leq -78\).
5. Interpret the result: the temperatures \(C\) on the surface of Mars lie between \(-90\) degrees Celsius and \(-78\) degrees Celsius, inclusive.

Key Concepts

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.