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Difficulty: EasyAbsolute Value Equations and Inequalities

For what values of the real number pp is the inequality −2∣p−4∣≥−10-2|p - 4| \geq -10 true?

  1. A
    p≤−1p \leq -1 or p≥9p \geq 9
  2. B
    −9≤p≤1-9 \leq p \leq 1
  3. −1≤p≤9-1 \leq p \leq 9Answer
  4. D
    p≤9p \leq 9
  5. E
    p≥9p \geq 9

Answer

−1≤p≤9-1 \leq p \leq 9
The correct answer is the inequality showing that pp is between −1-1 and 99, inclusive. First, divide both sides of the inequality −2∣p−4∣≥−10-2|p - 4| \geq -10 by −2-2 and reverse the inequality sign to obtain ∣p−4∣≤5|p - 4| \leq 5. Next, set up the compound inequality −5≤p−4≤5-5 \leq p - 4 \leq 5. Finally, add 44 to all parts of the inequality to isolate pp, which yields −1≤p≤9-1 \leq p \leq 9.

Step-by-Step Solution

1
Divide both sides of the inequality −2∣p−4∣≥−10-2|p - 4| \geq -10 by −2-2.
∣p−4∣≤5|p - 4| \leq 5
Dividing by a negative number reverses the direction of the inequality sign.
2
Rewrite the absolute value inequality ∣p−4∣≤5|p - 4| \leq 5 as a compound inequality.
−5≤p−4≤5-5 \leq p - 4 \leq 5
An inequality of the form ∣x∣≤a|x| \leq a for a≥0a \geq 0 is equivalent to −a≤x≤a-a \leq x \leq a.
3
Add 44 to all parts of the compound inequality to isolate pp.
−1≤p≤9-1 \leq p \leq 9
Adding a constant to all parts of an inequality preserves the inequality relationships and isolates the variable.

Key Concept

Solving absolute value inequalities involving multiplication or division by a negative number.
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Topics

Absolute Value Equations and Inequalities

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