Questions: Solve the following equation: -4x-9=9-3x

Solve the following equation:
-4x-9=9-3x
Transcript text: Solve the following equation: \[ -4 x-9=|9-3 x| \]
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Solution

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Solution Steps

To solve the equation \(-4x - 9 = |9 - 3x|\), we need to consider the definition of the absolute value function. This means we will split the equation into two cases: one where \(9 - 3x \geq 0\) and one where \(9 - 3x < 0\). We will then solve each case separately and check which solutions satisfy the original equation.

Step 1: Understand the Problem

We need to solve the equation: \[ -4x - 9 = |9 - 3x| \] This equation involves an absolute value, which means we need to consider both the positive and negative scenarios of the expression inside the absolute value.

Step 2: Set Up the Two Cases

The absolute value equation \(|9 - 3x|\) can be split into two cases:

  1. \(9 - 3x \geq 0\) which simplifies to \(9 - 3x\)
  2. \(9 - 3x < 0\) which simplifies to \(-(9 - 3x)\)
Step 3: Solve the First Case

For the first case, where \(9 - 3x \geq 0\): \[ -4x - 9 = 9 - 3x \] Solve for \(x\): \[ -4x - 9 = 9 - 3x \] Add \(3x\) to both sides: \[ -4x + 3x - 9 = 9 \] Simplify: \[ -x - 9 = 9 \] Add 9 to both sides: \[ -x = 18 \] Multiply both sides by -1: \[ x = -18 \]

Step 4: Check the Validity of the First Case

We need to check if \(x = -18\) satisfies the condition \(9 - 3x \geq 0\): \[ 9 - 3(-18) = 9 + 54 = 63 \geq 0 \] Since the condition is satisfied, \(x = -18\) is a valid solution for the first case.

Step 5: Solve the Second Case

For the second case, where \(9 - 3x < 0\): \[ -4x - 9 = -(9 - 3x) \] Simplify the right-hand side: \[ -4x - 9 = -9 + 3x \] Solve for \(x\): \[ -4x - 9 = -9 + 3x \] Add \(4x\) to both sides: \[ -9 = -9 + 7x \] Add 9 to both sides: \[ 0 = 7x \] Divide both sides by 7: \[ x = 0 \]

Step 6: Check the Validity of the Second Case

We need to check if \(x = 0\) satisfies the condition \(9 - 3x < 0\): \[ 9 - 3(0) = 9 \not< 0 \] Since the condition is not satisfied, \(x = 0\) is not a valid solution for the second case.

Final Answer

The only valid solution is: \[ \boxed{x = -18} \]

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