Find all real numbers x such that $ |5x-1| ≤ |3x-2|$

Question

I'm not entirely sure which proof method to use?

I'm stuck between induction or proof by cases.

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I have gotten a better understanding now, as I am using proof by cases, however it how can I prove that my answer is correct?

Answer 1

Squaring both side leads to: $(5x-1)^2-(3x-2)^2\leq 0 \iff (8x-3)(2x+1)\leq 0 \iff x\in [-\frac{1}{2};\frac{3}{8}]$

Answer 2

If you square it you get quadratic inequality: $$25x^2-10x+1\leq 9x^2-12x+4$$

Answer 3

Hint: Consider the cases $$5x-1\geq 0$$ $$3x-2\geq0$$ or $$5x-1\geq 0$$ and $$3x-2<0$$ or $$5x-1<0$$ and $$3x-2\geq 0$$ or $$5x-1<0$$ and $$3x-2<0$$

Answer 4

draw some pictures......................