Solve the inequality |2-3x| ≤ 11

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|2-3x| < 11

3|x-2/3| < 11

|x-2/3| < 11/3

The distance of x from 2/3 is less than 11/3.

<---O========+=========O--->

-9/3 2/3 13/3

-3 < x < 13/3

Hi Kay;

In the title, you explain this is <.

In the description, you explain this is ≤.

I will use the first...

|2-3x| < 11

Absolute value is always positive.

(2-3x) can be either...

+(2-3x) or -(2-3x).

When the absolute value of either is taken, it has the same result.

-(2-3x)<11

-2+3x<11

Let's add 2 to both sides...

-2+2+3x<11+2

3x<13

Divide both sides by 3...

(3x)/3<13/3

x<13/3

+(2-3x)<11

2-3x<11

Let's subtract 2 from both sides...

2-2-3x<11-2

-3x<9

Divide both sides by -3. Because this is a negative number < will switch to >...

(-3x)/-3>9/-3

x>-3

-3<x<13/3

Let's verify by plugging-in a few points within this domain...

|2-3x| < 11

x=0

|2-[(3)(0)]|<11

|2|<11

2<11-----YES

x=-1

|2-3x| < 11

|2-[(3)(-1)]|<11

|2+3|<11

5<11---YES

x=1

|2-3x| < 11

|2-[(3)(1)]|<11

|2-3|<11

-1<11-------YES

Let's verify by plugging-in a few points outside of this domain...

x=-4

|2-3x| < 11

|2-[(3)(-4)]|<11

|2+12|<11

13<11-----NO

x=5

|2-3x| < 11

|2-[(3)(5)]|<11

|2-15|<11

|-13|<11

13<11-------NO

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