A researcher is monitoring the temperature, T, in degrees Celsius, of a chemical reaction.… — SAT

A researcher is monitoring the temperature, $T$ , 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 $-5(T - 12) - 8(3 - 2T) < 14(T + 3)$ . Which of the following inequalities represents all possible values of $T$ for which the cooling system will activate?

A
$T < -2$
$T > -2$ Answer
C
$T > -42$
D
$T > 2$

Answer

The inequality representing all possible values of the temperature is

T > -2

The correct inequality is obtained by first expanding both sides of the inequality to get

-5T + 60 - 24 + 16T < 14T + 42

. Simplifying the left side yields

11T + 36 < 14T + 42

. Subtracting

14T

from both sides gives

-3T + 36 < 42

, and subtracting

36

from both sides gives

-3T < 6

. Finally, dividing both sides by

-3

and reversing the inequality sign results in

T > -2

Step-by-Step Solution

Distribute the constants on both sides of the inequality:

-5(T - 12) - 8(3 - 2T) < 14(T + 3)

-5T + 60 - 24 + 16T < 14T + 42

Remove the parentheses to prepare the inequality for simplification.

Combine the like terms on the left side of the inequality.

11T + 36 < 14T + 42

Simplify the expression before isolating the variable.

Subtract

14T

and

36

from both sides of the inequality.

-3T < 6

Group all variable terms on one side and constant terms on the other.

Divide both sides of the inequality by

-3

and reverse the inequality sign.

T > -2

Dividing by a negative number reverses the inequality relationship.

Key Concept

Solving multi-step linear inequalities in one variable, distributing coefficients, combining like terms, and reversing the inequality sign when dividing by a negative number.

Estimated Time:2m 0s

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