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Exercise 9 · Q3

Q.Solve: 4x−2<84x - 2 < 8, when x∈Zx \in \mathbb{Z}.

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✓ Free question

Solving 4x−2<84x-2<8 gives x<2.5x<2.5; restricted to integers, xx can be 22 or any smaller integer.

Solve a linear inequality by isolating xx; adding/subtracting or dividing by a positive number keeps the inequality direction unchanged. Then keep only values in the required set (here x∈Zx\in\mathbb{Z}).

  1. Given: 4x−2<84x - 2 < 8.
  2. Add 22 to both sides: 4x<104x < 10.
  3. Divide by 44 (positive): x<104=2.5x < \dfrac{10}{4} = 2.5.
  4. Restrict to integers: the integers less than 2.52.5 are {…,−2,−1,0,1,2}\{\dots,-2,-1,0,1,2\}, i.e. x≤2x \le 2 with x∈Zx\in\mathbb{Z}.
  5. Self-check: x=2⇒4(2)−2=6<8x=2\Rightarrow 4(2)-2=6<8 ✓; x=3⇒4(3)−2=10≮8x=3\Rightarrow 4(3)-2=10\not<8 ✗.
✓Final answer

x<2.5x < 2.5; for x∈Zx\in\mathbb{Z}, solution set ={…,−2,−1,0,1,2}=\{\dots,-2,-1,0,1,2\}.

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