What is the sum of all integer values of x that satisfy both |2x - 5| \le 9 and |x + 2| \g… — GRE General Test

What is the sum of all integer values of $x$ that satisfy both $|2x - 5| \le 9$ and $|x + 2| \ge 4$ ?

Answer: 27

The sum of all integer values of

x

that satisfy both inequalities is 27.

First, solving

|2x - 5| \le 9

yields

-9 \le 2x - 5 \le 9 \implies -4 \le 2x \le 14 \implies -2 \le x \le 7

. Second, solving

|x + 2| \ge 4

yields

x + 2 \ge 4 \implies x \ge 2

x + 2 \le -4 \implies x \le -6

. Taking the intersection of

-2 \le x \le 7

and

(x \ge 2 \text{ or } x \le -6)

gives the solution set

2 \le x \le 7

. The integer values satisfying this condition are 2, 3, 4, 5, 6, and 7. Summing these integers gives

2 + 3 + 4 + 5 + 6 + 7 = 27

Solve the inequality

|2x - 5| \le 9

-2 \le x \le 7

An inequality of the form

|u| \le a

(where

a \ge 0

) expands to

-a \le u \le a

. Adding 5 gives

-4 \le 2x \le 14

, and dividing by 2 yields

-2 \le x \le 7

Solve the inequality

|x + 2| \ge 4

x \ge 2 \text{ or } x \le -6

An inequality of the form

|u| \ge a

(where

a > 0

) expands to

u \ge a

u \le -a

. Subtracting 2 from both inequalities yields

x \ge 2

x \le -6

Determine the overlapping interval for both inequalities

2 \le x \le 7

Combining

-2 \le x \le 7

with

x \ge 2 \text{ or } x \le -6

eliminates

x \le -6

. The intersection of

[-2, 7]

and

[2, \infty)

[2, 7]

Identify the integer values in the solution interval and calculate their sum

The integers in the closed interval

[2, 7]

are 2, 3, 4, 5, 6, and 7. Adding them together gives

2 + 3 + 4 + 5 + 6 + 7 = 27

System of Linear Absolute Value Inequalities

Question