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Inequalities Involving Modulus Function

Number of int...

Question

Number of integer values of $x$ satisfying the inequality
$| x - 3 | + | 2 x + 4 | + | x | \leq 11$ is

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Solution

$| x - 3 | + | 2 x + 4 | + | x | \leq 11$
If $x \geq 3$ ,
$x - 3 + 2 x + 4 + x \leq 11$
$\Rightarrow 4 x \leq 10$
$\Rightarrow x \leq 2.5$ No solution

If $0 \leq x < 3$ ,
$- x + 3 + 2 x + 4 + x \leq 11 \Rightarrow x \leq 2 \Rightarrow x \in [0, 2]$

If $- 2 \leq x < 0$ ,
$- x + 3 + 2 x + 4 - x \leq 11$
$\Rightarrow 7 \leq 11$ which is true
$\Rightarrow x \in [- 2, 0)$

If $x < - 2$ ,
$- x + 3 - 2 x - 4 - x \leq 11$
$\Rightarrow - 4 x - 1 \leq 11$
$\Rightarrow x \geq - 3 \Rightarrow x \in [- 3, - 2)$

Hence, $x \in [0, 2] \cup [- 2, 0) \cup [- 3, - 2)$
i.e., $x \in [- 3, 2]$

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