What is the sum of all integer values of y that satisfy the inequality |2y - 3| \le 7? — ACT

What is the sum of all integer values of $y$ that satisfy the inequality $|2y - 3| \le 7$ ?

Answer: 12

The sum of all integer values of

y

that satisfy the inequality is 12.

Solving the inequality

|2y - 3| \le 7

requires setting up the compound inequality

-7 \le 2y - 3 \le 7

. Adding 3 to all parts gives

-4 \le 2y \le 10

, and dividing by 2 yields the interval

-2 \le y \le 5

. The integers in this closed interval are

-2, -1, 0, 1, 2, 3, 4

, and

5

. Summing these values gives 12, as the terms

-2

and

-1

cancel out with

2

and

1

Set up the compound inequality

-7 \le 2y - 3 \le 7

An absolute value inequality of the form

|a| \le b

translates to

-b \le a \le b

Isolate the term containing

y

-4 \le 2y \le 10

Add 3 to all three parts of the compound inequality to eliminate the

-3

Solve for

y

-2 \le y \le 5

Divide all three parts of the inequality by 2.

Identify the integer solutions in the interval

-2, -1, 0, 1, 2, 3, 4, 5

The inequality includes the endpoints, so the integers satisfying the inequality are all integers from

-2

through

5

, inclusive.

Calculate the sum of the integers

Adding the integers:

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 = 12

. The negative integers cancel out their corresponding positive counterparts (

-2

and

2

-1

and

1

Solving compound inequalities derived from absolute value inequalities and finding the sum of the integer solution set.

Estimated Time:1m 30s

Question