QQuestion
Solve the inequation, write the solution set and represent it on the real number line:
2x − \frac{5}{3} < \frac{3\mathrm{x}}{5} + 10 ≤ \frac{4\mathrm{x}}{5} + 11, x ∈ R.
✓Answer
Solving the left-hand part:
2x − \frac{5}{3} < \frac{3\mathrm{x}}{5} + 10
\frac{6\mathrm{x}-5}{3}<\frac{3\mathrm{x}+50}{5}Multiplying by 15:
5(6x − 5) < 3(3x + 50)
⇒ 30x − 25 < 9x + 150
⇒ 21x < 175
⇒ x < \frac{25}{3} … (1)
Solving the right-hand part:
\frac{3\mathrm{x}}{5} + 10 ≤ \frac{4\mathrm{x}}{5} + 11
⇒ 3x + 50 ≤ 4x + 55
⇒ x ≥ −5 … (2)
Combining (1) and (2):
−5 ≤ x < \frac{25}{3}
Therefore, the solution set is [−5, \frac{25}{3}).
On the number line, use a closed point at −5, an open point at \frac{25}{3}, and shade between them.
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