[tex]\frac{2 x+1}{4}\ \textgreater \ \frac{x+3}{3}[/tex]
Answer (1)
Multiply both sides by 12 to eliminate fractions: 4(x+3)"> 3 ( 2 x + 1 ) > 4 ( x + 3 ) .
Expand both sides: 4x + 12"> 6 x + 3 > 4 x + 12 .
Simplify and isolate x : 9"> 2 x > 9 .
Divide by 2 to find the solution: \frac{9}{2}"> x > 2 9 .
The solution to the inequality is \frac{9}{2}}"> x > 2 9 .
Explanation
Understanding the Inequality We are given the inequality \frac{x+3}{3}"> 4 2 x + 1 > 3 x + 3 . Our goal is to isolate x on one side of the inequality to find the solution set.
Eliminating Fractions To eliminate the fractions, we multiply both sides of the inequality by the least common multiple of 4 and 3, which is 12: 12 \cdot \frac{x+3}{3}"> 12 ⋅ 4 2 x + 1 > 12 ⋅ 3 x + 3 Simplifying, we get: 4(x+3)"> 3 ( 2 x + 1 ) > 4 ( x + 3 )
Expanding the Inequality Next, we distribute the constants on both sides of the inequality: 4x + 12"> 6 x + 3 > 4 x + 12
Isolating x Now, we want to isolate x . Subtract 4 x from both sides: 4x - 4x + 12"> 6 x − 4 x + 3 > 4 x − 4 x + 12 12"> 2 x + 3 > 12
Further Isolating x Subtract 3 from both sides: 12 - 3"> 2 x + 3 − 3 > 12 − 3 9"> 2 x > 9
Solving for x Finally, divide both sides by 2: \frac{9}{2}"> 2 2 x > 2 9 \frac{9}{2}"> x > 2 9 So the solution to the inequality is \frac{9}{2}"> x > 2 9 , which is equivalent to 4.5"> x > 4.5 .
Final Answer The solution to the inequality \frac{x+3}{3}"> 4 2 x + 1 > 3 x + 3 is \frac{9}{2}"> x > 2 9 .
Examples
Imagine you're comparing two different cell phone plans. Plan A costs 4 2 x + 1 dollars per month, where x is the number of gigabytes of data you use. Plan B costs 3 x + 3 dollars per month for the same amount of data. Solving the inequality \frac{x+3}{3}"> 4 2 x + 1 > 3 x + 3 tells you for what data usage (in gigabytes) Plan A is more expensive than Plan B. This helps you decide which plan is more economical based on your data needs. Understanding and solving linear inequalities is crucial for making informed decisions in various real-life scenarios, from budgeting to comparing costs.
