Problem 52 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|4 x+3|=|9-2 x|\)

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -6\).

Step 1: Understand the Concept of Absolute Values

The given equation is \(|4x + 3| = |9 - 2x|\). This means the absolute value of \(4x + 3\) is equal to the absolute value of \(9 - 2x\). Therefore, we need to consider both the positive and negative cases of each side.

Step 2: Case 1 - Both Expressions Positive

Assume both sides are positive. Thus, we can drop the absolute value bars and solve the equation directly: \[4x + 3 = 9 - 2x\] Add \(2x\) to both sides to get: \[6x + 3 = 9\]Subtract 3 from both sides: \[6x = 6\] Divide both sides by 6: \[x = 1\]

Step 3: Case 2 - First Expression Negative, Second Positive

Assume \(4x + 3\) is negative and \(9 - 2x\) is positive. This gives us:\[-(4x + 3) = 9 - 2x\] This simplifies to: \[-4x - 3 = 9 - 2x\] Add \(4x\) to both sides: \[-3 = 9 + 2x\] Subtract 9 from both sides: \[-12 = 2x\] Divide both sides by 2: \[x = -6\]

Step 4: Case 3 - First Expression Positive, Second Negative

Assume \(4x + 3\) is positive and \(9 - 2x\) is negative. This gives us:\[4x + 3 = -(9 - 2x)\] Expand and simplify: \[4x + 3 = -9 + 2x\] Subtract \(2x\) from both sides: \[2x + 3 = -9\] Subtract 3 from both sides: \[2x = -12\] Divide both sides by 2: \[x = -6\]

Step 5: Case 4 - Both Expressions Negative

Assume both sides are negative:\[-(4x + 3) = -(9 - 2x)\] This simplifies to:\[-4x - 3 = -9 + 2x\] Add \(4x\) to both sides: \[-3 = -9 + 6x\] Add 9 to both sides: \[6 = 6x\] Divide both sides by 6: \[x = 1\]

Step 6: Verify All Solutions

The solutions found are \(x = 1\) and \(x = -6\). Substitute these back into the original equation to check:For \(x = 1\): \(|4(1) + 3| = |9 - 2(1)|\) translates to \(|7| = |7|\), which is true.For \(x = -6\): \(|4(-6) + 3| = |9 - 2(-6)|\) translates to \(|-21| = |21|\), which is also true.

Key Concepts

Absolute Value

The concept of absolute value is essential in understanding certain types of equations. Essentially, an absolute value measures the distance of a number from zero on a number line. It's always expressed as a non-negative number. This means regardless of whether the original number is positive or negative, its absolute value will always be positive.

• For example, both \( |5| \) and \( |-5| \) are equal to 5 because the distance from zero for both numbers is the same.
• Similarly, \( |0| \) is zero, as zero has no distance from itself.

When dealing with absolute value equations like \(|4x + 3| = |9 - 2x|\), it is important to recognize that the distance on both sides of the equation from zero must be equal. This necessitates considering multiple possible cases: both expressions being positive, one being negative, or both being negative.

Equation Solving

Solving an equation involves finding the values of the variable that make the equation true. For absolute value equations, we must carefully explore different scenarios since absolute values can be interpreted in various ways.
In our problem \(|4x+3|=|9-2x|\), the approach to solve it effectively involves:

• Setting each absolute value expression equal directly: \( 4x+3 = 9-2x \).
• Considering cases where one expression is negative while the other is positive like \( -(4x+3) = 9-2x \).
• Exploring the inverse where \(4x+3\) is positive and \(9-2x\) is negative.
• Finally, examining both expressions when negative: \(-(4x+3) = -(9-2x)\).

By solving each case separately, we determine which values of \(x\) satisfy the entire equation. Remember, once potential solutions are calculated, always verify by substituting back into the original equation to confirm their correctness.

Algebra Concepts

The understanding of algebra concepts is crucial when tackling equations, especially those involving absolute values. Key algebraic steps are employed when breaking down absolute value equations:

• Think of absolute value as a two-way street: each expression inside \(| \ |\) has both a positive and negative equation associated with it.
• Use algebraic principles such as adding, subtracting, or dividing both sides of an equation to isolate the variable \(x\).

For instance, in \(4x + 3 = 9 - 2x\), algebra requires us to combine like terms or reverse operations to simplify the equation:

• To isolate \(x\), one could add or subtract terms symmetrically from both sides.
• After simplifying, division may be necessary to solve for \(x\).

In summary, strong algebra skills empower you to manipulate and solve equations systematically. Applying these tried-and-tested procedures aids in achieving correct solutions consistently.