solve this inequalities 1/3x-2 not <2/3x-6

Question

Inequalities

Answer 1

Given inequality 1/3x-2 not < 2/3x-6

I assume the inequality symbol is >.

1/(3x-2) > 2/(3x-6)

Multiple to each side by 3x-2.

(1/3x-2)3x-2 > (2/3x-6)3x-2

1 > 2(3x-2)/3x-6

Multiple to each side by 3x-6.

1(3x-6) > 2(3x-2)(3x-6)/3x-6

3x-6 > 2(3x-2)

3x-6 > 6x-4

-6 > 6x-4-3x

-6+4 > 3x

-2 > 3x

-2/3 > x

x < -2/3

Solution x <-2/3.

Comments

The solution of the inequality is

x  < -2/3 

2/3 <  x  < 2

---

The symbol  "not <"(not less than) means "≥"(greater than or equal).

Answer 2

I assume the inequality symbol is >.

The rational inequality numarator critical value is x  = -2/3

The rational  inequality denominator critical values at x  = 2/3 x  = 2

Step 2

Our critical points make partition the x - axis into four intervals.

Pick a point on each interval.

x  = -2 , x  = 0 , x  = 1 and x  = 3

Now compute 

The rational inequality

f(-2)

f(0)

f(1)

f(3)

f (-2) < 0 f (x) will be negitive all x in the interval (-∞,-2),Similarly since  f (0) > 0 f (x ) will be positive (-2/3,2/3), Since f (1) < 0 f (x ) will be negitive all x  in the interval (2/3,2),Since f (3) > 0 f (x ) will be positive for all x interval (2,∞)

we indicate this on the x - axis by inserting plus or minus signs on the x - axis. I use color coding instead: green for positive, pink for negative:

 The rational inequality

Solution

x  < -2/3 

2/3 <  x  < 2

Comments

The symbol  "not <"(not less than) means "≥"(greater than or equal).

Answer 3

The inequatity is 1/3 x - 2 not < 2/3x - 6.

1/3 x - 2 is not less than 2/3x - 6. That means 1/3 x - 2 is greater than or equal to 2/3x - 6.

1/3 x - 2 ≥ 2/3 x - 6

1/3 x - 2 ≥ 2/3 x - 6

Apply Subtraction Property of Inequality: If a ≥ b then a - c ≥ b - c.

Subtract 2/3 x from each side.

1/3 x - 2 - 2/3 x ≥ 2/3 x - 6 - 2/3 x

Group like terms.

1/3 x - 2/3 x - 2 ≥ 2/3 x - 2/3 x - 6

Combine like terms.

(1-2)/3 x - 2 ≥ 0 - 6

-1/3 x - 2 ≥ - 6

Apply Additon Property of Inequality: If a ≥ b then a + c ≥ b + c.

Add 2 to each side.

-1/3 x - 2 + 2 ≥ - 6 + 2

-1/3 x - 0 ≥ - 4

-1/3 x  ≥ - 4

Apply Multiplication Property of Inequality: If a ≥ b then a*-c ≤ b*-c.

(-1/3 x) (-3)  ≤ (- 4) (-3)

x ≤ 12

The answer is x ≤ 12.