Problem 43 - Elementary and Intermediate Algebra

Question

Solve each equation. \(-5|2 x-9|+14=14\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{9}{2}\).

Step 1: Isolate the Absolute Value Expression

Start by simplifying the equation \(-5|2x - 9| + 14 = 14\). Subtract 14 from both sides to isolate the absolute value term: \(-5|2x - 9| = 0\).

Step 2: Solve for the Absolute Value

Next, divide both sides by -5 to solve for the absolute value expression: \(|2x - 9| = 0\).

Step 3: Solve the Equation Inside the Absolute Value

Since \(|2x - 9| = 0\), set the expression inside the absolute value equal to 0: \(2x - 9 = 0\).

Step 4: Solve for x

Add 9 to both sides to get \(2x = 9\). Then, divide both sides by 2 to solve for \(x\): \(x = \frac{9}{2}\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements formed by setting two algebraic expressions equal to one another. These equations can include variables, numbers, and operational symbols. Solving algebraic equations involves finding the value of the variable that makes the equation true. To achieve this, we perform various operations that keep the equation balanced.
In the equation \(-5|2x-9| +14 = 14\), we see an example of an algebraic equation that incorporates an absolute value. Here, the task is to solve for the unknown variable \(x\) within the absolute value setup. It is crucial for students to recognize the role of algebraic equations in creating mathematical models to solve real-world problems, and learning to manipulate these equations is a fundamental skill in algebra.

Isolate the Absolute Value

When solving equations involving absolute values, the first step is often to isolate the absolute value expression. This helps in simplifying the equation considerably and allows us to handle the absolute value separately.
In our example, \(-5|2x-9| +14 = 14\), the absolute value expression \(|2x-9|\) must be isolated. Subtracting 14 from both sides simplifies the equation to \(-5|2x - 9| = 0\). Then, we divide each side of the equation by -5, which gives us \(|2x - 9| = 0\).
Absolute value equations set to zero, like \(|2x - 9| = 0\), indicate that what is inside the absolute value must also be zero. This is because the absolute value of any number except zero cannot be zero.

Solve for x

After isolating and simplifying the absolute value expression, the next step is solving the equation contained within the absolute value for the variable \(x\). For \(|2x - 9| = 0\), remove the absolute value signs because the only number whose absolute value is zero is zero itself. This gives us \(2x - 9 = 0\).
Solving \(2x - 9 = 0\), we first add 9 to both sides to get \(2x = 9\). Following this, divide both sides by 2, resulting in \(x = \frac{9}{2}\).
It's essential to perform operations that keep the equation balanced, meaning whatever you do to one side must be done to the other. Understanding this concept is key to correctly solving for \(x\) and verifying that your solution satisfies the original equation.