For all real values of n that satisfy the inequality \frac{3 - 2n}{4} < \frac{3n - 22}{5},… — ACT

For all real values of $n$ that satisfy the inequality $\frac{3 - 2n}{4} < \frac{3n - 22}{5}$ , what is the smallest possible integer value of $n$ ?

Cevap: 5

The smallest integer value of

n

that satisfies the inequality is 5.

Solving the inequality yields

n > \frac{103}{22} \approx 4.68

. The smallest integer greater than

4.68

5

Multiply both sides by 20 to clear the denominators.

5(3 - 2n) < 4(3n - 22)

To eliminate the fractions and simplify the inequality.

Distribute the constants on both sides.

15 - 10n < 12n - 88

To remove the parentheses.

Subtract

12n

from both sides.

15 - 22n < -88

To group the terms containing the variable on the left side.

Subtract 15 from both sides.

-22n < -103

To isolate the variable term on the left side.

Divide both sides by

-22

and reverse the inequality sign.

n > \frac{103}{22}

Dividing by a negative number reverses the direction of the inequality sign.

Convert the fraction to a decimal to identify the boundary.

n > 4.68

To find the smallest integer value that satisfies this condition.

Identify the smallest integer greater than

4.68

5

The smallest integer greater than

4.68

5

Solving linear inequalities by applying inverse operations and reversing the inequality sign when multiplying or dividing by a negative number.

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