Textbook Question
The temperatures on the surface of Mars in degrees Celsiusapproximately satisfy the inequality |C + 84 | ≤ 6. What range of temperaturescorresponds to this inequality?
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1. Equations & Inequalities
Linear Inequalities
3:55 minutesProblem 83
Textbook Question
In Exercises 59–94, solve each absolute value inequality.- 4|1 - x| < - 16
Verified step by step guidance1
Step 1: Start by isolating the absolute value expression. Divide both sides of the inequality by -4. Remember, dividing by a negative number reverses the inequality sign. The inequality becomes: .
Step 2: Recall the definition of absolute value inequalities. For , the inequality splits into two cases: or . Apply this to the inequality: or .
Step 3: Solve each case separately. For the first case, , subtract 1 from both sides to isolate : . Then divide by -1 (reversing the inequality sign): .
Step 4: For the second case, , subtract 1 from both sides to isolate : . Then divide by -1 (reversing the inequality sign): .
Step 5: Combine the solutions from both cases. The solution to the inequality is or . This represents the values of that satisfy the original inequality.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value
Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving inequalities that involve expressions within these bars.
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Inequalities
Inequalities express a relationship where one side is not equal to the other, using symbols like <, >, ≤, or ≥. In the context of absolute value inequalities, they indicate the range of values that satisfy the condition. For instance, solving |x| < a means finding all x values that are within a distance a from zero.
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Properties of Inequalities
When manipulating inequalities, certain properties must be observed, such as the fact that multiplying or dividing by a negative number reverses the inequality sign. This is essential when isolating variables in absolute value inequalities. Understanding these properties helps ensure that the solutions derived from the inequalities are valid.
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