Grade 12Algebra

If x satisfies |3x – 2| + |3x – 4| + |3x – 6| >= 12 to what interval does x belong? |...| represents modulus function. [BITSAT 2013]

Raghav

7 Years agoGrade 12

1 Answer

Samyak Jain

7 Years ago

|3x – 2| + |3x – 4| + |3x – 6| $\geq$ 12.

First we find critical values of x by equating each modulus expression to zero.

Thus critical values of x are 2/3 , 4/3 and 6/3 = 2.

Case 1. x $\geq$ 2 ...(1)

3x – 2 + 3x – 4 + 3x – 6 $\geq$ 12 $\Rightarrow$ 9x – 12 $\geq$ 12

x $\geq$ 24 / 9 ...(2)

Taking intersection of (1) and (2), we get x $\geq$ 24 / 9 ...(i)

Case 2. 4/3 $\leq$ x

3x – 2 + 3x – 4 – (3x – 6) $\geq$ 12 $\Rightarrow$ 3x $\geq$ 12

x $\geq$ 4 ...(4)

Intersection of (3) and (4) gives x $\epsilon$ $\phi$ ...(ii)

Case 3. 2/3 $\leq$ x

3x – 2 – (3x – 4) – (3x – 6) $\geq$ 12 $\Rightarrow$ – 3x + 8 $\geq$ 12 $\Rightarrow$ 3x $\leq$ – 4

x $\leq$ – 4/3 ...(6)

Taking intersection of (5) and (6), we get x $\epsilon$ $\phi$ ...(iii)

Case 4. x

– (3x – 2) – (3x – 4) – (3x – 6) $\geq$ 12 $\Rightarrow$ – 9x + 12 $\geq$ 12

x $\leq$ 0 ...(8)

Intersection of (7) and (8) gives x $\leq$ 0 ...(iv)

Union of (i), (ii), (iii) & (iv) gives x $\geq$ 24/9 $\cup$ x $\leq$ 0 i.e. x $\epsilon$ (– $\infty$ , 0] $\cup$ [24 / 9 , $\infty$ ).

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If x satisfies |3x – 2| + |3x – 4| + |3x – 6| >= 12 to what interval does x belong? |...| represents modulus function. [BITSAT 2013]

Raghav

1 Answer

Samyak Jain

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