Problem 69 - College Algebra

Question

Solve each absolute value equation or indicate that the equation has no solution. $$7|5 x|+2-16$$

Step-by-Step Solution

Answer

The solutions to the given absolute value equation are \(x = 0.4\) and \(x = -0.4\).

Step 1: Isolate the Absolute Value

Subtract 2 from both sides of the equation to isolate the absolute value: \(7|5x| = 16-2\), which simplifies to \(7|5x| = 14\)

Step 2: Further Isolation

Further isolate the absolute value by dividing both sides of the equation by 7: \(|5x| = 14 / 7\), which simplifies to \(|5x| = 2\).

Step 3: Handle Possible Cases

Because the absolute value of a number can be positive or negative, there are two possible cases: when 5x is positive and when 5x is negative. For case 1, \(5x = 2\). To solve for x, divide both sides of this equation by 5, which gives \(x = 2 / 5\) or \(x = 0.4\). For case 2, \(5x = -2\). Dividing both sides by 5 gives \(x = -2 / 5\) or \(x = -0.4\).

Key Concepts

Isolating the Absolute Value

To isolate the absolute value, start by rearranging the equation so that the absolute value expression is all by itself on one side of the equation. For the exercise at hand, this involves moving other terms away from the absolute value term.
First, look at the equation: \(7|5x|+2=16\). We need to remove the constant next to the absolute value term.

• Subtract 2 from both sides: \(7|5x|=14\).
• Now divide both sides by 7 to get \(|5x|=2\).

Algebraically isolating the absolute value simplifies subsequent steps, allowing for two potential equations to resolve.
Remember, you want the format \(|Ax|=B\). This lays the groundwork to consider the two cases for 'x'. Understanding why and how to isolate the absolute value is crucial because it streamlines the solving process.

Solving the Equation

After isolating the absolute value, solving the equation involves considering two scenarios. This stems from the inherent property of absolute value: \(|a|\) is the distance from zero, thus could be 'a' or '-a'.

• First, set up \(5x=2\). For this, divide both sides by 5, giving \(x=0.4\).
• Second, handle \(5x=-2\). Also divide both sides by 5, which gives \(x=-0.4\).

Both solutions need checking as solving absolute value equations often reveals one solution, two solutions, or sometimes none. Always substitute back to the original equation to ensure correctness.
Dual possibilities define how absolute values encode both ends of the spectrum. This breaks down to solving linear equations but with extra care for potential outcomes.

Understanding Algebraic Expressions

Algebraic expressions are combinations of constants, variables and operations. In these situations, breaking them down efficiently is essential.
Consider each part: in our exercise, the expression \(7|5x|+2\) consists of multiplication, an absolute value, and addition. Here, recognizing that multiplication and addition adhere to the order of operations.

• The term \(7|5x|\) means the value inside the absolute value expression is multiplied by 7.'
• The addition of 2 helps understand what gets isolated or moved during adjustments.

Algebraic expressions guide you through manipulations needed to isolate and solve. They require clarity between numeric and variable manipulations; recognizing this strives towards solving broader mathematical challenges.
Crafty algebraic handling can make or break understanding and solving equations, particularly those involving absolute values.