To solve the inequality \(\frac{(x+2)}{4} - \frac{(2x-3)}{3} < 4\), we will follow a step-by-step approach to isolate \(x\) and find the range of values that satisfy the inequality. ### Step 1: Combine the fractions First, we need to combine the two fractions on the left side of the inequality. To do this, we need a common denominator. The denominators are 4 and 3, and the least common multiple (LCM) of 4 and 3 is 12. We will rewrite each fraction with a denominator of 12: \[ \frac{x+2}{4} = \frac{3(x+2)}{12} = \frac{3x + 6}{12} \] \[ \frac{2x-3}{3} = \frac{4(2x-3)}{12} = \frac{8x - 12}{12} \] Now we can rewrite the inequality: \[ \frac{3x + 6}{12} - \frac{8x - 12}{12} < 4 \] ### Step 2: Combine the fractions Now that both fractions have the same denominator, we can combine them: \[ \frac{3x + 6 - (8x - 12)}{12} < 4 \] Distributing the negative sign: \[ \frac{3x + 6 - 8x + 12}{12} < 4 \] Combine like terms in the numerator: \[ \frac{-5x + 18}{12} < 4 \] ### Step 3: Eliminate the fraction To eliminate the fraction, multiply both sides of the inequality by 12 (note that since 12 is positive, the direction of the inequality does not change): \[ -5x + 18 < 48 \] ### Step 4: Isolate \(x\) Now, we will isolate \(x\) by first subtracting 18 from both sides: \[ -5x < 48 - 18 \] \[ -5x < 30 \] Next, divide both sides by -5. Remember that when dividing by a negative number, we must reverse the inequality sign: \[ x > -6 \] ### Conclusion The solution to the inequality is \(x > -6\). ### Explanation of Options Now, let's analyze the provided options: - **A. \(x > -6\)**: This is the correct answer, as we derived it from the inequality. - **B. \(x > -3\)**: This option is incorrect because it is a more restrictive condition than \(x > -6\). While all values greater than -3 are also greater than -6, not all values greater than -6 are greater than -3. - **C. \(x < 8\)**: This option is irrelevant to our solution. The inequality does not provide any upper limit for \(x\), so we cannot conclude that \(x\) must be less than 8. - **D. \(x < 4\)**: Similar to option C, this option is also irrelevant. The inequality does not impose any upper limit on \(x\). ### Revision Summary - To solve the inequality, combine fractions using a common denominator. - Isolate the variable by eliminating fractions and rearranging the inequality. - Remember to reverse the inequality sign when dividing by a negative number. - The correct solution is \(x > -6\), which corresponds to option A. This thorough approach ensures that you understand each step and the reasoning behind the solution.