If x is a real number that satisfies both of the inequalities |3x + 4| \ge 11 and |x - 1| … — GRE General Test

If $x$ is a real number that satisfies both of the inequalities $|3x + 4| \ge 11$ and $|x - 1| < 6$ , which of the following could be the value of $x$ ? Select all that apply.

A
$-5$
B
$-3$
$3$ Cevap
$5$ Cevap
E
$7$

Cevap

The values 3 and 5 satisfy both inequalities.

Solving the first inequality

|3x + 4| \ge 11

yields

x \le -5

x \ge \frac{7}{3}

. Solving the second inequality

|x - 1| < 6

yields

-5 < x < 7

. Intersecting these two regions, the interval

x \le -5

does not overlap with

-5 < x < 7

because

-5

is excluded from the second inequality. The overlap occurs only for

\frac{7}{3} \le x < 7

. Among the choices, 3 and 5 fall within this valid interval.

Adım Adım Çözüm

Solve the absolute value inequality

|3x + 4| \ge 11

x \le -5

x \ge \frac{7}{3}

An absolute value inequality of the form

|u| \ge k

(where

k > 0

) splits into two separate inequalities:

u \ge k

u \le -k

. Solving

3x + 4 \ge 11

gives

3x \ge 7 \implies x \ge \frac{7}{3}

. Solving

3x + 4 \le -11

gives

3x \le -15 \implies x \le -5

Solve the absolute value inequality

|x - 1| < 6

-5 < x < 7

An absolute value inequality of the form

|u| < k

is equivalent to the compound inequality

-k < u < k

. Thus,

-6 < x - 1 < 6

. Adding 1 to all parts yields

-5 < x < 7

Determine the intersection of the two solution sets.

\frac{7}{3} \le x < 7

The portion

x \le -5

has no overlap with

-5 < x < 7

because

-5

is excluded by the strict inequality in the second condition. The portion

x \ge \frac{7}{3}

overlaps with

-5 < x < 7

to yield the interval

[\frac{7}{3}, 7)

Evaluate which of the given options fall inside

[\frac{7}{3}, 7)

The numbers 3 and 5 belong to the interval, while

-5

-3

, and

7

do not.

Since

\frac{7}{3} \approx 2.33

, the values 3 and 5 fall strictly between 2.33 and 7.

Anahtar Kavram

Solving systems of linear absolute value inequalities by finding the intersection of compound solution intervals

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Cevap

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