What are the possible solutions for the inequality |2x-1|+|x+2|\geq 4x?

[thereddevils]

Homework Statement

$$|2x-1|+|x+2|\geq 4x$$

Homework Equations

The Attempt at a Solution

For x<-2 , $$-(2x-1)-(x+2)\geq 4x$$

$$x\leq -\frac{1}{7}$$

For $$x\geq \frac{1}{2}$$

$$2x-1+x+2\geq 4x$$

$$x\leq 1$$

For $$-2 \leq x < \frac{1}{2}$$ ,

$$-(2x-1)+x+2\geq 4x$$

$$x\leq \frac{3}{5}$$

after combining , the solution would be $$x\leq -\frac{1}{7}$$

AM i correct ? but the answer given is $$x\leq 1$$

 

[tiny-tim]

Hi thereddevils!

Your three answers are correct, but you're not putting them together correctly.

For example, you have:

if x ≤ -2, it's true for x ≤ -1/7.

so it's true for all x ≤ -2.

ok, now try the others. ​

 

[thereddevils]

Hi thereddevils!

Your three answers are correct, but you're not putting them together correctly.

For example, you have:

if x ≤ -2, it's true for x ≤ -1/7.

so it's true for all x ≤ -2.

ok, now try the others. ​

thanks tiny , yeah , i am confused with the last part , what i did is to put x<= -1/7 , x<=1 , x<= 3/5 on the number line and take the intersection which is what i got .

so say for x>= 1/2 , x can be 4 , x<=1 , so 4<=1 ?? this is not true

i am sorry , i still do not understand the last part .

 

[tiny-tim]

There are three possibilities:

x ≤ -2, -2 ≤ x ≤ 1/2, 1/2 ≤ x.

call them A B and C.

So A or B or C.

If A, the equation is true if (say) a.

If B, the equation is true if (say) b.

If C, the equation is true if (say) c.

So the equation is true if (A and a) or (B and b) or (C and c).