Problem 27 - Algebra for College Students

Question

Solve each equation. $$ |0.02 x-1|=2.50 $$

Step-by-Step Solution

Answer

The solutions are \x = 175\ and \x = -75\.

Step 1: Understand the absolute value equation

The given equation is \(|0.02x - 1| = 2.50\). This means that the expression inside the absolute value can be either 2.50 or -2.50, because the absolute value measures the distance from zero.

Step 2: Create two separate equations

Set up two equations to solve for \( x \): \(0.02x - 1 = 2.50\) and \(0.02x - 1 = -2.50\).

Step 3: Solve the first equation

Solve \(0.02x - 1 = 2.50\): \ \begin{align*} 0.02x - 1 &= 2.50 \ 0.02x &= 3.50 \ x &= \frac{3.50}{0.02} \ x &= 175 \end{align*} \

Step 4: Solve the second equation

Solve \(0.02x - 1 = -2.50\): \ \begin{align*} 0.02x - 1 &= -2.50 \ 0.02x &= -1.50 \ x &= \frac{-1.50}{0.02} \ x &= -75 \end{align*} \

Step 5: Combine the results

The solutions to the equation \(|0.02x - 1| = 2.50\) are \x = 175\ and \x = -75\.

Key Concepts

Solving Equations

When solving equations, the main goal is to find the value of the variable that makes the equation true. For example, in our exercise, the equation is \(|0.02 x-1|=2.50\). \
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To solve this, we need to remove the absolute value and set up two different equations. Absolute values can be tricky because they measure the distance from zero, so we always need to consider both the positive and negative cases. \
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• First, recognize the absolute value equation given.
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• Break down the absolute value equation into two separate equations.
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• Solve each of these equations individually to find the values of \( x \).
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\ By understanding these steps, we bridge the gap from seeing the problem to solving it efficiently.

Absolute Value

The absolute value of a number is its distance from zero on the number line. It’s always non-negative. For example, the absolute value of both 3 and -3 is 3 because both are three units away from zero. \
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In algebra, absolute value equations such as \(|0.02 x-1|=2.50\) come up frequently. Here's how to tackle them: \

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• Understand that \(|0.02 x-1|=2.50\) or \(|0.02 x-1|=-2.50\).
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• Create two different scenarios: \(0.02 x-1 = 2.50\) and \(0.02 x-1 = -2.50\).
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• Solve each of these separately for \(x\).
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\ Absolute value helps simplify equations by breaking them down to their core components, making them easier to solve.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In this exercise, the expression \(0.02x - 1\) inside the absolute value is what we focus on. \
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Here’s how to handle algebraic expressions step by step: \* First, isolate the absolute value expression. \* Next, split it into two equations, removing the absolute value. \* Finally, solve each equation for \(x\). \
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For example, in \(0.02x - 1 = 2.50\): \

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• Add 1 to both sides to isolate the term with \( x \). This gives \(0.02x = 3.50 \).
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• Divide by 0.02, resulting in \( x = 175 \).
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• Repeat similar steps for the negative scenario: \(0.02x - 1 = -2.50 \), and after solving, we get \( x = -75 \).
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\ By understanding algebraic expressions, we can navigate through equations more easily and find the values of \( x \) that satisfy the conditions.