Examkin

Soru

Zorluk: ZorAbsolute Value Equations and Inequalities

For all real values of tt that satisfy the inequality 12−4∣3−2t∣≤−812 - 4|3 - 2t| \leq -8, which of the following inequalities represents the complete set of possible values of tt?

  1. A
    t≤−1t \leq -1
  2. B
    −1≤t≤4-1 \leq t \leq 4
  3. C
    t≥−1t \geq -1 or t≤4t \leq 4
  4. t≤−1t \leq -1 or t≥4t \geq 4Cevap
  5. E
    t≤1t \leq 1 or t≥2t \geq 2

Cevap

t≤−1t \leq -1 or t≥4t \geq 4
The correct answer is t≤−1t \leq -1 or t≥4t \geq 4. To solve the inequality, we first subtract 12 from both sides to get −4∣3−2t∣≤−20-4|3 - 2t| \leq -20. Next, dividing both sides by −4-4 and reversing the inequality sign gives ∣3−2t∣≥5|3 - 2t| \geq 5. This absolute value inequality splits into two cases: 3−2t≥53 - 2t \geq 5 or 3−2t≤−53 - 2t \leq -5. Solving the first case, we subtract 3 to get −2t≥2-2t \geq 2, and dividing by −2-2 while reversing the inequality sign yields t≤−1t \leq -1. Solving the second case, we subtract 3 to get −2t≤−8-2t \leq -8, and dividing by −2-2 while reversing the inequality sign yields t≥4t \geq 4. Combining these, we obtain the solution set t≤−1t \leq -1 or t≥4t \geq 4.

Adım Adım Çözüm

1
Isolate the absolute value term by subtracting 12 from both sides of the inequality.
−4∣3−2t∣≤−20-4|3 - 2t| \leq -20
Before splitting an absolute value inequality, the absolute value expression must be isolated on one side.
2
Divide both sides of the inequality by −4-4 and reverse the inequality sign.
∣3−2t∣≥5|3 - 2t| \geq 5
Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.
3
Split the absolute value inequality ∣u∣≥c|u| \geq c (where c>0c > 0) into two separate inequalities: u≥cu \geq c or u≤−cu \leq -c.
3−2t≥53 - 2t \geq 5 or 3−2t≤−53 - 2t \leq -5
An absolute value inequality of the form ∣u∣≥c|u| \geq c represents values that are at least cc units away from zero, which lie in two disjoint intervals.
4
Solve the first inequality: 3−2t≥53 - 2t \geq 5.
−2t≥2  ⟹  t≤−1-2t \geq 2 \implies t \leq -1
Subtracting 3 from both sides gives −2t≥2-2t \geq 2. Dividing by −2-2 and reversing the inequality sign yields t≤−1t \leq -1.
5
Solve the second inequality: 3−2t≤−53 - 2t \leq -5.
−2t≤−8  ⟹  t≥4-2t \leq -8 \implies t \geq 4
Subtracting 3 from both sides gives −2t≤−8-2t \leq -8. Dividing by −2-2 and reversing the inequality sign yields t≥4t \geq 4.
6
Combine the solutions from both cases to express the complete solution set.
t≤−1t \leq -1 or t≥4t \geq 4
The complete solution set is the union of the solutions to both cases.

Anahtar Kavram

Solving absolute value inequalities of the form ∣ax+b∣≥c|ax + b| \geq c by translating them into compound inequalities and carefully reversing the inequality sign when multiplying or dividing by negative numbers.
Bu soruyu puanla

Konular

Absolute Value Equations and Inequalities

Bu konudan daha fazla soru