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Zorluk: ZorLinear Inequalities in One Variable

A researcher is monitoring the temperature, TT, in degrees Celsius, of a chemical reaction. The target temperature range is maintained by an automated cooling system that activates when the temperature satisfies the inequality −5(T−12)−8(3−2T)<14(T+3)-5(T - 12) - 8(3 - 2T) < 14(T + 3). Which of the following inequalities represents all possible values of TT for which the cooling system will activate?

  1. A
    T<−2T < -2
  2. T>−2T > -2Cevap
  3. C
    T>−42T > -42
  4. D
    T>2T > 2

Cevap

The inequality representing all possible values of the temperature is T>−2T > -2.
The correct inequality is obtained by first expanding both sides of the inequality to get −5T+60−24+16T<14T+42-5T + 60 - 24 + 16T < 14T + 42. Simplifying the left side yields 11T+36<14T+4211T + 36 < 14T + 42. Subtracting 14T14T from both sides gives −3T+36<42-3T + 36 < 42, and subtracting 3636 from both sides gives −3T<6-3T < 6. Finally, dividing both sides by −3-3 and reversing the inequality sign results in T>−2T > -2.

Adım Adım Çözüm

1
Distribute the constants on both sides of the inequality: −5(T−12)−8(3−2T)<14(T+3)-5(T - 12) - 8(3 - 2T) < 14(T + 3).
−5T+60−24+16T<14T+42-5T + 60 - 24 + 16T < 14T + 42
Remove the parentheses to prepare the inequality for simplification.
2
Combine the like terms on the left side of the inequality.
11T+36<14T+4211T + 36 < 14T + 42
Simplify the expression before isolating the variable.
3
Subtract 14T14T and 3636 from both sides of the inequality.
−3T<6-3T < 6
Group all variable terms on one side and constant terms on the other.
4
Divide both sides of the inequality by −3-3 and reverse the inequality sign.
T>−2T > -2
Dividing by a negative number reverses the inequality relationship.

Anahtar Kavram

Solving multi-step linear inequalities in one variable, distributing coefficients, combining like terms, and reversing the inequality sign when dividing by a negative number.
Tahmini Süre:2m 0s
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Konular

Linear Inequalities in One Variable

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