QQuestion
Solve the following inequation, write down the solution set and represent it on the real number line:
−3 + x ≤ \frac{7x}{2} + 2 < 8 + 2x, x ∈ I.
✓Answer
Given:
−3 + x ≤ \frac{7x}{2} + 2 < 8 + 2x
Solving the left-hand part:
−3 + x ≤ \frac{7x}{2} + 2
⇒ \frac{7x}{2} − x ≥ −3 − 2
⇒ \frac{7x-2x}{2} ≥ −5
⇒ \frac{5x}{2} ≥ −5
⇒ 5x ≥ −10
⇒ x ≥ −2 … (1)
Solving the right-hand part:
\frac{7x}{2} + 2 < 8 + 2x
⇒ \frac{7x}{2} − 2x < 8 − 2
⇒ \frac{7x-4x}{2} < 6
⇒ \frac{3x}{2} < 6
⇒ 3x < 12
⇒ x < 4 … (2)
Combining (1) and (2):
−2 ≤ x < 4
Since x ∈ I, the solution set is:
{−2, −1, 0, 1, 2, 3}
On the number line, mark filled points at −2, −1, 0, 1, 2 and 3.
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