Problem 78 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ |x-2|+|x-3|+2|x-4|=9 $$

Step-by-Step Solution

Answer

The two solutions to the given equation are \(x = 1\) and \(x = 4.5\).

Step 1: Consider signs of expressions for this case

For this case, the following conditions apply: - x-2<0 - x-3<0 - x-4<0

Step 2: Rewrite the equation with appropriate signs

As all the expressions are negative, we need to multiply each expression inside the absolute values by -1. The equation becomes: \[ -(x-2) - (x-3) - 2(x-4) = 9 \]

Step 3: Simplify the equation

Expand and simplify the equation: \[ -x + 2 - x + 3 - 2x + 8 = 9 \] Combine like terms and solve for x: \[ -4x + 13 = 9 \]

Step 4: Solve for x

Subtract 13 from both sides and divide by -4: \[ x = \frac{13 - 9}{4} = 1 \] Since x is within the specified range (x<2, x<3, x<4) this is a valid solution: \(x=1\). #Case 2: 2<=x<3, x<4#

Step 5: Consider signs of expressions for this case

For this case, the following conditions apply: - x-2>=0 - x-3<0 - x-4<0

Step 6: Rewrite the equation with appropriate signs

The equation becomes: \[ (x-2) - (x-3) - 2(x-4) = 9 \]

Step 7: Simplify the equation

Simplify the equation: \[ -1 - 2x + 8 = 9 \]

Step 8: Solve for x

Subtract 8 from both sides and divide by -2: \[ x = \frac{1}{2} \] This solution is not within the specified range (2<=x<3), so we discard it. #Case 3: 3<=x<4#

Step 9: Consider signs of expressions for this case

For this case, the following conditions apply: - x-2>=0 - x-3>=0 - x-4<0

Step 10: Rewrite the equation with appropriate signs

The equation becomes: \[ (x-2) + (x-3) - 2(x-4) = 9 \]

Step 11: Simplify the equation

Simplify the equation: \[ 2x - 5 - 2x + 8 = 9 \]

Step 12: Solve for x

Since 2x and -2x cancel each other out, we are left with: \[ 3 = 9 \] This equation is not possible, so there is no solution within this range. #Case 4: x>=4#

Step 13: Consider signs of expressions for this case

For this case, the following conditions apply: - x-2>=0 - x-3>=0 - x-4>=0

Step 14: Rewrite the equation with appropriate signs

The equation becomes: \[ (x-2) + (x-3) + 2(x-4) = 9 \]

Step 15: Simplify the equation

Simplify the equation: \[ x + x + 2x - 9 = 9 \]

Step 16: Solve for x

Combine like terms and solve for x: \[ 4x = 18 \] Divide by 4: \[ x = \frac{18}{4} = 4.5 \] Since x is within the specified range (x>=4), this is a valid solution: \(x=4.5\). The two solutions to the equation are \(x = 1\) and \(x = 4.5\).

Key Concepts

Piecewise Functions

Piecewise functions are a fundamental concept when dealing with absolute value expressions. Absolute value describes how far a number is from zero, disregarding whether it’s positive or negative. But, when equations involve absolute values, like \[|x-2| + |x-3| + 2|x-4| = 9\]we often need a piecewise approach. This is because absolute values impact the equation differently based on whether the expressions inside them are negative or positive.

• Think of a piecewise function as a recipe with different steps depending on specific conditions.
• The conditions help decide whether to flip the sign or not.
• This process helps in examining all the possible scenarios.

In our example, you break down the equation into sections based on ranges of *x*, checking each range to rewrite the equation without the absolute value symbol. Each piece corresponds to specific intervals where the equation behaves differently, letting you solve the problems neatly step-by-step.

Equation Solving

Once the piecewise analysis is performed, Equation solving becomes crucial. Solving equations typically involves isolating a variable, usually *x*, until it stands alone. In the context of absolute value equations, rewritten without absolute signs, equations might look more standard, such as: \[-(x-2) - (x-3) - 2(x-4) = 9\]This process involves:

• Rewriting the entire equation based on the conditions identified.
• Simplifying by performing operations across each term—this often involves combining like terms.
• Reassessing conditions to ensure the solution fits the initial operational assumptions based on the range of *x*.

For each piecewise segment, this technique can reveal whether a valid solution exists within the specified range. If the simplification does not logically result in a valid statement, the condition’s segment can be discarded.

Mathematical Problem Solving

Mathematical Problem Solving is the overarching skill that brings together understanding and methodologies, such as piecewise functions and solving equations. When tackling complex equations involving absolute values, structured problem solving offers several advantages:

• It ensures all potential solutions are considered by evaluating every possible scenario or case.
• The process requires a robust comprehension of mathematics—from evaluating conditions, simplifying expressions, to logically reviewing potential answers.
• This strategic layout safeguards against overlooking valid solutions or accepting invalid ones, ensuring comprehensive verification against initial conditions.

Thus, effective problem-solving apparatuses like these not only resolve the given equation but also strengthen your understanding of mathematical principles, enhancing your recognition of patterns and application in various contexts. Mathematical problem solving isn't just about finding the right answer from calculations, but refining an approach that's replicable across different math challenges.