What is the smallest integer value of y that satisfies the inequality 5(2 - y) < 3(y - 6)? — ACT

What is the smallest integer value of $y$ that satisfies the inequality $5(2 - y) < 3(y - 6)$ ?

Answer: 4

The smallest integer value of

y

that satisfies the inequality is

4

Evaluating the inequality leads to

y > 3.5

. The smallest integer greater than

3.5

4

. Substituting

y = 4

into the original inequality gives

5(2 - 4) < 3(4 - 6) \implies -10 < -6

, which is true. Substituting the next smallest integer,

3

, gives

-5 < -9

, which is false.

Distribute the coefficients to the terms inside the parentheses on both sides.

10 - 5y < 3y - 18

To clear the parentheses and simplify the terms.

Subtract

3y

and

10

from both sides of the inequality to group the variable terms on one side and constants on the other.

-8y < -28

To isolate the variable term.

Divide both sides by

-8

and reverse the inequality sign because of division by a negative number.

y > 3.5

To solve the inequality for

y

Identify the smallest integer that satisfies the inequality

y > 3.5

4

To find the smallest integer value greater than

3.5

Solving multi-step linear inequalities involving distribution and division by a negative number.

Instead of subtracting variables to the left, we can add

5y

to both sides to keep the variable coefficient positive:

10 - 5y < 3y - 18 \implies 10 < 8y - 18 \implies 28 < 8y \implies y > 3.5

. This avoids the need to divide by a negative number and flip the sign, reducing the risk of a sign-flip error.

Estimated Time:1m 0s

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