Does the inequality |x+3|+|x-3|+|x|< 3 have any solution.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The absolute value of a number x , |x| is equal to x if x `>=` 0 and it is equal to –x if x<0.

For the given inequality|x+3|+|x-3|+|x|< 3:

Assume x lies in the set {`oo` , -3}

The inequality is equivalent to -(x+3) - (x – 3) - x < 3

=> -x - 3 - x + 3 - x < 3

=> -3x < 3

=> x > -1

But it has been assumed that x lies in {`oo` , -3}. There is no solution of x in the given set.

Assume x lies in the set [-3, 0}

The inequality is equivalent to (x+3) - (x – 3) - x < 3

=> x + 3 - x + 3 - x < 3

=> 6 - x < 3

=> x > 3

But it has been assumed that x lies in [-3,0}. There is no solution of x in the given set.

Assume x lies in the set [0, 3}

The inequality is equivalent to (x+3) - (x – 3) + x < 3

=> x + 3 - x + 3 + x < 3

=> 6 + x < 3

=> x < -3

But it has been assumed that x lies in [0, 3}. There is no solution of x in the given set.

Assume x lies in the set [3, ` oo`}

The inequality is equivalent to (x+3) + (x – 3) + x < 3

=> x + 3 + x - 3 + x < 3

=> 3x  < 3

=> x < 1

But it has been assumed that x lies in [3, `oo` }. There is no solution of x in the given set.

The sets of values considered above include all the values of x. As no solution for x is found in any of the sets, the given inequality has no solution.

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