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If 3x + 22x ≥ 5x, then the solution set for x is:
  • a)
    (-∞, 2]
  • b)
    [2, ∞)
  • c)
    [0, 2]
  • d)
    {2}
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If 3x + 22x ≥ 5x, then the solution set for x is:a)(-∞, 2]b)[...
We have,
3x + 22x ≥ 5x
⇒ (3/5)x + (4/5)x ≥ 1
⇒ (3/5)x + (4/5)x ≥ (3/5)2 + (4/5)2
⇒ x ≤ 2 ⇒ x ∈ (-∞, 2]
[If ax + bx ≥ 1 and a2 + b2 = 1], then x ∈ (-∞, 2)
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Community Answer
If 3x + 22x ≥ 5x, then the solution set for x is:a)(-∞, 2]b)[...
Understanding the Inequality
To solve the inequality 3x + 22x ≥ 5x, we first simplify the left side:
- Combine like terms: 3x + 22x = 25x.
- The inequality then becomes: 25x ≥ 5x.
Simplifying the Inequality
Next, we isolate x:
- Subtract 5x from both sides: 25x - 5x ≥ 0.
- This simplifies to: 20x ≥ 0.
Finding the Solution Set
Now, we can solve for x:
- Divide both sides by 20: x ≥ 0.
This means that x can be any value greater than or equal to 0.
Interpreting the Solution Set
The solution set consists of all real numbers starting from 0 and extending to positive infinity. Therefore, we can express this solution set in interval notation:
- The interval is [0, +∞).
Conclusion
Given the options provided:
- a) [0, ∞) - This is correct.
- b) (-∞, 2] - Incorrect as it includes negative values.
- c) [0, 2] - Incorrect as it limits the upper bound.
- d) (-2, ∞) - Incorrect as it includes negative values.
Thus, the correct answer is option 'A':
[0, ∞).
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