Questions: Solve the following equation:
-4x-9=9-3x
Transcript text: Solve the following equation:
\[
-4 x-9=|9-3 x|
\]
Solution
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:
\(9 - 3x \geq 0\) which simplifies to \(9 - 3x\)
\(9 - 3x < 0\) 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.