On the real number line, x is a real number such that its distance from 1 is at most 6, an… — GRE General Test

On the real number line, $x$ is a real number such that its distance from $1$ is at most $6$ , and its distance from $-2$ is at least $4$ . Which of the following inequalities represents all possible values of $x$ ?

$2 \le x \le 7$ Cevap
B
$-5 \le x \le 7$
C
$-6 \le x \le 7$
D
$-5 \le x \le 2$
E
$-5 \le x \le -2 \text{ or } 2 \le x \le 7$

Cevap

The inequality

2 \le x \le 7

represents all possible values of

x

The distance condition 'at most 6 units from 1' translates to

|x - 1| \le 6

, yielding the interval

[-5, 7]

. The condition 'at least 4 units from -2' translates to

|x + 2| \ge 4

, yielding

x \le -6

x \ge 2

. Finding the values that belong to both conditions requires taking the intersection of

[-5, 7]

and

(-\infty, -6] \cup [2, \infty)

. Since

[-5, 7]

does not overlap with

(-\infty, -6]

, the only overlapping region is

[2, 7]

, which corresponds to

2 \le x \le 7

Adım Adım Çözüm

Express the geometric distance conditions as absolute value inequalities

Condition 1:

|x - 1| \le 6

. Condition 2:

|x - (-2)| = |x + 2| \ge 4

The distance between two numbers

a

and

b

on the number line is given by

|a - b|

Solve the first inequality

|x - 1| \le 6

-6 \le x - 1 \le 6 \implies -5 \le x \le 7

An absolute value inequality of the form

|u| \le k

(for

k \ge 0

) unwraps to

-k \le u \le k

Solve the second inequality

|x + 2| \ge 4

x + 2 \ge 4

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

x \le -6

An absolute value inequality of the form

|u| \ge k

(for

k \ge 0

) unwraps to

u \ge k

u \le -k

Find the intersection of the two solution sets

Combine

[-5, 7]

with

(-\infty, -6] \cup [2, \infty)

. The intersection of

[-5, 7]

and

(-\infty, -6]

is empty since

-5 > -6

. The intersection of

[-5, 7]

and

[2, \infty)

[2, 7]

, which means

2 \le x \le 7

The variable

x

must satisfy both conditions simultaneously.

Anahtar Kavram

Distance on a number line expressed as absolute value inequalities and finding overlapping intervals

Tahmini Süre:2m 0s

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Cevap

Adım Adım Çözüm

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