GPT
To solve the inequality:
2x−33x−5≥3\frac{2x - 3}{3x - 5} \geq 3
Step 1: Rearrange the inequality
Move all terms to one side:
2x−33x−5−3≥0\frac{2x - 3}{3x - 5} - 3 \geq 0
Combine the terms under a common denominator:
2x−33x−5−3(3x−5)3x−5≥0\frac{2x - 3}{3x - 5} - \frac{3(3x - 5)}{3x - 5} \geq 0
Simplify the numerator:
2x−3−9x+153x−5≥0\frac{2x - 3 - 9x + 15}{3x - 5} \geq 0
Combine like terms:
−7x+123x−5≥0\frac{-7x + 12}{3x - 5} \geq 0
Step 2: Find critical points
The inequality changes sign at the points where the numerator or denominator is zero. Solve for these critical points:
- **Numerator:**−7x+12=0 ⟹ x=127-7x + 12 = 0 \implies x = \frac{12}{7}
- **Denominator:**3x−5=0 ⟹ x=533x - 5 = 0 \implies x = \frac{5}{3}
Thus, the critical points are x=127x = \frac{12}{7} and x=53x = \frac{5}{3}.
Step 3: Test intervals between critical points
The critical points divide the number line into three intervals: (−∞,127)(-\infty, \frac{12}{7}), (127,53)(\frac{12}{7}, \frac{5}{3}), and (53,∞)(\frac{5}{3}, \infty). Test the sign of the inequality in each interval.
Interval 1: (−∞,127)(-\infty, \frac{12}{7})
Choose x=0x = 0:
−7(0)+123(0)−5=12−5=−125<0\frac{-7(0) + 12}{3(0) - 5} = \frac{12}{-5} = -\frac{12}{5} < 0
The inequality is not satisfied.
Interval 2: (127,53)(\frac{12}{7}, \frac{5}{3})
Choose x=2x = 2:
−7(2)+123(2)−5=−14+126−5=−21=−2<0\frac{-7(2) + 12}{3(2) - 5} = \frac{-14 + 12}{6 - 5} = \frac{-2}{1} = -2 < 0
The inequality is not satisfied.
Apologies for the interruption in the previous response. Let me carefully continue solving the inequality.
We were analyzing the intervals, and it seems there was an error in choosing x=2x = 2, which lies outside the interval (127,53)\left( \frac{12}{7}, \frac{5}{3} \right). Let’s properly analyze the intervals:
3. Test intervals between critical points
Interval 1: (−∞,127)(-\infty, \frac{12}{7})
Choose x=0x = 0:
−7(0)+123(0)−5=12−5=−125<0\frac{-7(0) + 12}{3(0) - 5} = \frac{12}{-5} = -\frac{12}{5} < 0
The inequality is not satisfied.
Interval 2: (127,53)\left( \frac{12}{7}, \frac{5}{3} \right)
Choose x=1x = 1:
−7(1)+123(1)−5=−7+123−5=5−2=−52<0\frac{-7(1) + 12}{3(1) - 5} = \frac{-7 + 12}{3 - 5} = \frac{5}{-2} = -\frac{5}{2} < 0
The inequality is not satisfied.
Interval 3: (53,∞)(\frac{5}{3}, \infty)
Choose x=2x = 2:
−7(2)+123(2)−5=−14+126−5=−21=−2\frac{-7(2) + 12}{3(2) - 5} = \frac{-14 + 12}{6 - 5} = \frac{-2}{1} = -2
It seems I made some mistakes in step 3 earlier. Let’s carefully revisit the solution and properly analyze the intervals to ensure clarity.
The inequality to solve is:
−7x+123x−5≥0\frac{-7x + 12}{3x - 5} \geq 0
Critical points:
- From the numerator: −7x+12=0 ⟹ x=127-7x + 12 = 0 \implies x = \frac{12}{7}.
- From the denominator: 3x−5=0 ⟹ x=533x - 5 = 0 \implies x = \frac{5}{3}.
These divide the number line into intervals: (−∞,127)(-\infty, \frac{12}{7}), (127,53)(\frac{12}{7}, \frac{5}{3}), and (53,∞)(\frac{5}{3}, \infty).
Interval Analysis:
Interval 1: (−∞,127)(-\infty, \frac{12}{7})
Choose x=0x = 0:
−7(0)+123(0)−5=12−5=−125\frac{-7(0) + 12}{3(0) - 5} = \frac{12}{-5} = -\frac{12}{5}
The result is negative, so the inequality is not satisfied in this interval.
Interval 2: (127,53)\left( \frac{12}{7}, \frac{5}{3} \right)
Choose x=1x = 1:
−7(1)+123(1)−5=−7+123−5=5−2=−52\frac{-7(1) + 12}{3(1) - 5} = \frac{-7 + 12}{3 - 5} = \frac{5}{-2} = -\frac{5}{2}
The result is negative, so the inequality is not satisfied in this interval.
Interval 3: (53,∞)(\frac{5}{3}, \infty)
Choose x=2x = 2:
−7(2)+123(2)−5=−14+126−5=−21=−2\frac{-7(2) + 12}{3(2) - 5} = \frac{-14 + 12}{6 - 5} = \frac{-2}{1} = -2
The result is also negative, so the inequality is not satisfied in this interval.