3(4x-3)=4|x-(4x-3)|

Question

3(4x-3)=4|x-(4x-3)|

Bytelearn · Accepted Answer

Distribute and Simplify: First, let's simplify the left side of the equation by distributing the 3 into the parentheses. 3(4x-3) = 12x - 9Simplified Equation: Now, we have the simplified equation: 12x - 9 = 4|x - (4x - 3)|Simplify Absolute Value: Next, we need to simplify the expression inside the absolute value on the right side of the equation. |x - (4x - 3)| = |x - 4x + 3|Further Simplification: Simplify the expression inside the absolute value further. |x - 4x + 3| = |-3x + 3|Consider Two Cases: Now, we have the equation: 12x - 9 = 4|-3x + 3| We need to consider two cases for the absolute value: one where the expression inside is positive, and one where it is negative.Case 1 Inequality: Case 1: The expression inside the absolute value is positive, i.e., -3x + 3 ≥ 0. We solve for x in this inequality to find the range of x for which this case applies. -3x + 3 ≥ 0 -3x ≥ -3 x ≤ 1Solve Case 1 Equation: Now, we solve the equation for Case 1, assuming -3x + 3 is positive. 12x - 9 = 4(-3x + 3)Combine Like Terms: Distribute the 4 on the right side of the equation. 12x - 9 = -12x + 12Isolate x Term: Add 12x to both sides of the equation to get all x terms on one side. 12x + 12x - 9 = -12x + 12x + 12 24x - 9 = 12Solve for x: Add 9 to both sides of the equation to isolate the term with x. 24x - 9 + 9 = 12 + 9 24x = 21Check Validity: Divide both sides by 24 to solve for x. x = 21 / 24 x = 7/8Case 2 Inequality: Now we need to check if x = 7/8 satisfies the inequality x ≤ 1 for Case 1. 7/8 ≤ 1 is true, so x = 7/8 is a valid solution for Case 1.Solve Case 2 Equation: Case 2: The expression inside the absolute value is negative, i.e., -3x + 3 < 0. We solve for x in this inequality to find the range of x for which this case applies. -3x + 3 < 0 -3x < -3 x > 1Remove Double Negative: Now, we solve the equation for Case 2, assuming -3x + 3 is negative. 12x - 9 = 4(-(-3x + 3))Distribute and Simplify: Simplify the right side of the equation by removing the double negative. 12x - 9 = 4(3x - 3)Subtract Like Terms: Distribute the 4 on the right side of the equation. 12x - 9 = 12x - 12Contradiction - No Solution: Subtract 12x from both sides of the equation to get all x terms on one side. 12x - 12x - 9 = 12x - 12x - 12 0 - 9 = -12Simplify the equation. -9 = -12 This is a contradiction, which means there is no solution for x in Case 2.

$3(4 x-3)=4|x-(4 x-3)|$

Full solution

More problems from Evaluate absolute value expressions

Related topics

3(4x−3)=4∣x−(4x−3)∣ 3(4 x-3)=4|x-(4 x-3)| 3(4x−3)=4∣x−(4x−3)∣

$3(4 x-3)=4|x-(4 x-3)|$