Problem 95 - Intermediate Algebra

Question

Solve each equation. $$ 6-3|10 x+5|=6 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\).

Step 1: Simplify the Equation

Start by isolating the absolute value expression. In the equation \( 6 - 3|10x + 5| = 6 \), subtract 6 from both sides to get \( -3|10x + 5| = 0 \).

Step 2: Divide to Isolate the Absolute Value

Divide both sides by -3 to isolate the absolute value expression: \(|10x + 5| = 0\).

Step 3: Solve the Absolute Value Equation

If an absolute value is equal to zero, the expression inside must also be zero. Therefore, set the expression \(10x + 5\) equal to zero: \(10x + 5 = 0\).

Step 4: Solve for \(x\)

Solve the equation \(10x + 5 = 0\). Subtract 5 from both sides: \(10x = -5\). Then divide both sides by 10 to find \(x = -\frac{1}{2}\).

Key Concepts

Step-by-Step Solutions

Solving absolute value equations can seem tricky, but breaking down the process into manageable steps makes it easier. Let's dive into the specific approach taken in solving an equation like the one from our exercise.

The equation given was 6 - 3|10x + 5| = 6. Start by simplifying it. First, isolate the absolute value expression by removing constants outside of it. This is done by subtracting 6 from both sides, leading to -3|10x + 5| = 0.

• Always start by isolating the absolute value on one side of the equation.
• Use inverse operations to balance the equation. For instance, here we subtracted 6.

Next, divide both sides by -3 to make the absolute value expression alone. Now it reads |10x + 5| = 0. This is a crucial step as it prepares the equation for solving absolute values.

Finally, knowing that an absolute value equating to zero implies the expression inside must be zero, solve 10x + 5 = 0 by isolating x. Subtract 5 and then divide by 10 to get x = -\( \frac{1}{2} \).
This approach ensures clarity and accuracy.

Intermediate Algebra

Intermediate algebra is a level of mathematics that bridges basic arithmetic and introductory algebra to more advanced topics. It's essential for understanding more complex equations and mathematical concepts.

At this stage, you'll encounter concepts like rational expressions, quadratic equations, and absolute value equations. These topics require a solid understanding of operations with numbers and variables.

• Manipulating equations, such as isolating variables, is a key skill.
• Understanding properties of operations (like distribution, subtraction, and division) helps in transforming and solving equations efficiently.

In the problem we solved, dividing both sides of the equation by -3 and setting 10x + 5 to zero is a prime example of intermediate algebra. It's at this stage that you start to recognize patterns in equations and solutions, enhancing your problem-solving skills.

Absolute Value Properties

Absolute value refers to the distance of a number from zero on a number line, always expressed as a non-negative number. When solving equations involving absolute values, understanding their properties is crucial.

• The absolute value of a number is always positive, except for zero, whose absolute value is zero.
• If |A| = B, then A could be B or -B, but when B is zero, A must be zero too.

This property was particularly useful in our example because as soon as we reached the equation |10x + 5| = 0, we knew 10x + 5 must also be zero. Solving for x gave our desired result.

These properties allow you to convert absolute value equations into more straightforward linear equations, advancing your solving capabilities in algebra.