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Algebra Level 3

25 x + 177 = ∣ 11 x − 33 ∣ \large 25x + 177 = |11x-33 |

Find all the real numbers x x satisfying the equation above.

Notation : ∣ ⋅ ∣ | \cdot | denotes the absolute value function .

x = − 15 x= -15 x = − 15 , − 4 x=-15,-4 There is no solution x = − 4 x=-4

William Lindsey by William Lindsey , 35, USA

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2 solutions

William Lindsey
Jun 17, 2016

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Sam Bealing
Jun 19, 2016

Consider the following two cases:

Case 1: x ≥ 3 ⟹ 11 x − 33 ≥ 0 ⟹ ∣ 11 x − 33 ∣ = 11 x − 33 x \geq 3 \implies 11x-33 \geq 0 \implies \left |11x-33 \right |=11x-33

25 x + 177 = 11 x − 33 ⟹ 14 x = − 210 ⟹ x = − 15 25x+177=11x-33 \implies 14x=-210 \implies x=-15

This contradicts x ≥ 3 x \geq 3 so there are no solutions.

Case 2: x ≤ 3 ⟹ 11 x − 33 ≤ 0 ⟹ ∣ 11 x − 33 ∣ = 33 − 11 x x \leq 3 \implies 11x-33 \leq 0 \implies \left |11x-33 \right |=33-11x

25 x + 177 = 33 − 11 x ⟹ 36 x = − 144 ⟹ x = − 4 25x+177=33-11x \implies 36x=-144 \implies x=-4

This meets the condition of x ≤ 3 x \leq 3 so is a valid solution.

x = − 4 \color{#20A900}{\boxed{\boxed{x=-4}}}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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