Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x > 2 \)
Solution
Solve the equation \( 1+\frac{x-5}{3}-2x<-4 \).
Solve the inequality by following steps:
- step0: Solve for \(x\):
\(1+\frac{x-5}{3}-2x<-4\)
- step1: Multiply both sides:
\(\left(1+\frac{x-5}{3}-2x\right)\times 3<-4\times 3\)
- step2: Multiply the terms:
\(3+x-5-6x<-12\)
- step3: Simplify:
\(-2-5x<-12\)
- step4: Move the constant to the right side:
\(-5x<-12+2\)
- step5: Add the numbers:
\(-5x<-10\)
- step6: Change the signs:
\(5x>10\)
- step7: Divide both sides:
\(\frac{5x}{5}>\frac{10}{5}\)
- step8: Divide the numbers:
\(x>2\)
The solution to the inequality \(1+\frac{x-5}{3}-2x<-4\) is \(x>2\).
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Simplify this solution Extra Insights
To solve the inequality \( 1+\frac{x-5}{3}-2 x<-4 \), we start by simplifying the expression. First, isolate the fraction: \[ \frac{x-5}{3} - 2x + 1 < -4 \] Next, subtract 1 from both sides: \[ \frac{x-5}{3} - 2x < -5 \] Now, multiply the whole inequality by 3 to eliminate the fraction (remembering that multiplying by a positive number does not change the direction of the inequality): \[ x - 5 - 6x < -15 \] Combine like terms: \[ -5x - 5 < -15 \] Add 5 to both sides: \[ -5x < -10 \] Now, divide by -5 and remember to flip the inequality symbol: \[ x > 2 \] Thus, the solution to the inequality is: \[ \boxed{x > 2} \]
