Examkin

Soru

Zorluk: OrtaLinear Inequalities and Absolute Value

If xx is a real number that satisfies both ∣3−x∣≤5|3 - x| \le 5 and 2x−8−3≤−2\frac{2x - 8}{-3} \le -2, which of the following could be the value of xx? Indicate all such values.

  1. A
    −1-1
  2. B
    44
  3. 77Cevap
  4. 7.57.5Cevap
  5. 88Cevap

Cevap

The values 77, 7.57.5, and 88 satisfy both inequalities.
Solving ∣3−x∣≤5|3 - x| \le 5 gives −2≤x≤8-2 \le x \le 8. Solving 2x−8−3≤−2\frac{2x - 8}{-3} \le -2 by multiplying by −3-3 and reversing the inequality sign gives 2x−8≥62x - 8 \ge 6, or x≥7x \ge 7. Taking the intersection of both conditions yields 7≤x≤87 \le x \le 8. Among the choices, the values 77, 7.57.5, and 88 fall within this range.

Adım Adım Çözüm

1
Solve the absolute value inequality ∣3−x∣≤5|3 - x| \le 5.
−2≤x≤8-2 \le x \le 8
Rewrite as a compound inequality −5≤3−x≤5-5 \le 3 - x \le 5. Subtracting 33 gives −8≤−x≤2-8 \le -x \le 2. Multiplying by −1-1 and reversing inequality signs yields −2≤x≤8-2 \le x \le 8.
2
Solve the linear inequality 2x−8−3≤−2\frac{2x - 8}{-3} \le -2.
x≥7x \ge 7
Multiply both sides by −3-3, making sure to flip the inequality sign: 2x−8≥62x - 8 \ge 6. Adding 88 yields 2x≥142x \ge 14, so x≥7x \ge 7.
3
Find the intersection of the two solution sets.
7≤x≤87 \le x \le 8
Combining −2≤x≤8-2 \le x \le 8 and x≥7x \ge 7 gives the range 7≤x≤87 \le x \le 8.
4
Test the given options against the combined range 7≤x≤87 \le x \le 8.
The values 77, 7.57.5, and 88 fall within [7,8][7, 8], while −1-1 and 44 do not.
Only numbers greater than or equal to 77 and less than or equal to 88 satisfy both conditions.

Anahtar Kavram

Solving compound linear and absolute value inequalities, ensuring inequality signs are reversed when multiplying or dividing by negative numbers.
Bu soruyu puanla

Konular

Linear Inequalities and Absolute Value

Bu konudan daha fazla soru