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Zorluk: Çok zorLinear Inequalities and Absolute Value

Which of the following values of xx satisfy the inequality ∣2x−1∣≤5−x|2x - 1| \le 5 - x? Select all that apply.

  1. −4-4Cevap
  2. −1-1Cevap
  3. 11Cevap
  4. D
    33
  5. E
    −5-5

Cevap

The values of xx that satisfy the inequality are −4-4, −1-1, and 11.
Solving the compound inequality −(5−x)≤2x−1≤5−x-(5 - x) \le 2x - 1 \le 5 - x yields the solution range −4≤x≤2-4 \le x \le 2. The candidate values −4-4, −1-1, and 11 all lie within this range, making them valid solutions.

Adım Adım Çözüm

1
Determine the non-negativity constraint for the right-hand side expression.
Since an absolute value ∣2x−1∣|2x - 1| must be non-negative, we must have 5−x≥05 - x \ge 0, which simplifies to x≤5x \le 5.
An absolute value cannot be less than a negative number.
2
Rewrite the absolute value inequality ∣2x−1∣≤5−x|2x - 1| \le 5 - x as a compound linear inequality.
−(5−x)≤2x−1≤5−x-(5 - x) \le 2x - 1 \le 5 - x, which expands to −5+x≤2x−1≤5−x-5 + x \le 2x - 1 \le 5 - x.
For any non-negative expression BB, ∣A∣≤B|A| \le B is logically equivalent to −B≤A≤B-B \le A \le B.
3
Solve the left-hand inequality −5+x≤2x−1-5 + x \le 2x - 1.
Subtracting xx from both sides gives −5≤x−1-5 \le x - 1. Adding 11 to both sides yields x≥−4x \ge -4.
Isolating xx establishes the lower bound of the solution set.
4
Solve the right-hand inequality 2x−1≤5−x2x - 1 \le 5 - x.
Adding xx to both sides gives 3x−1≤53x - 1 \le 5. Adding 11 yields 3x≤63x \le 6, so x≤2x \le 2.
Isolating xx establishes the upper bound of the solution set.
5
Intersect all constraints to find the valid domain for xx.
Combining x≥−4x \ge -4, x≤2x \le 2, and x≤5x \le 5 gives the closed interval [−4,2][-4, 2].
A valid value of xx must satisfy all component inequalities simultaneously.
6
Evaluate the candidate choices against the solution interval [−4,2][-4, 2].
The values −4-4, −1-1, and 11 fall within [−4,2][-4, 2], whereas 33 and −5-5 fall outside this interval.
Only numbers inside [−4,2][-4, 2] satisfy the original inequality.

Anahtar Kavram

Solving Absolute Value Inequalities with Variable Expressions
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Konular

Linear Inequalities and Absolute Value

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