Problem 75 - College Algebra

Question

Solve absolute value inequality. \(\left|\frac{2 x+2}{4}\right| \geq 2\)

Step-by-Step Solution

Answer

The solution to the absolute value inequality is \(x \leq -5\) or \(x \geq 3\).

Step 1: Setup the inequality without the absolute value

Treat the expression within the absolute value symbols as a positive quantity and setup the first inequality as \(\frac{2 x+2}{4} \geq 2\).

Step 2: Simplify the inequality

Multiply both sides by 4 to clear the fraction: \(2x + 2 \geq 8\). Then simplify by subtracting 2 from both sides: \(2x \geq 6\). Lastly, divide both sides by 2 to isolate x: \(x \geq 3\).

Step 3: Setup the inequality considering the absolute value as a negative quantity

This time, treat the expression within the absolute value symbols as a negative quantity, which gives us \(-\frac{2 x+2}{4} \geq 2\).

Step 4: Simplify the inequality

Multiply both sides by 4 to clear the fraction: \(-2x - 2 \geq 8\). Simplify by adding 2 to both sides: \(-2x \geq 10\). Finally, divide both sides by -2 to isolate x. Don't forget to switch the inequality sign when multiplying or dividing by a negative number: \(x \leq -5\).

Key Concepts

Inequality Solving

Inequality solving involves finding the values of variables that make an inequality true. Think of it as similar to solving an equation, but with a twist! Inequalities use symbols like "<" or ">", which means we are not looking for exact values but rather a range of possible values.
When working with absolute value inequalities, remember that absolute value represents the distance from zero, so it is always positive. Therefore, you often have to break the inequality into two separate cases: one for the positive scenario and another for the negative.

• First, solve the inequality normally as if the expression inside the absolute value is positive.
• Next, solve it as if the expression inside is negative, remembering to flip the inequality sign when dealing with negatives.

Practicing these steps will build your confidence in solving inequalities efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, and division. They are the building blocks of algebra and help describe mathematical relationships.
In the context of our exercise, the expression \( \frac{2x+2}{4} \) includes a fraction. Fractions in algebra can initially look intimidating, but they can be simplified or eliminated by performing operations on both sides of the equation.

• To eliminate the fraction, multiply every term by the denominator. This clears the fraction and makes the expression easier to work with.
• When simplifying algebraic expressions, apply basic operations, such as distributing and combining like terms, carefully.

Simplifying these expressions correctly is essential in solving any algebra problem, particularly inequalities.

College Algebra

College algebra builds upon concepts learned in high school, providing more complex problems and applications. It includes solving equations, inequalities, and understanding functions.
Absolute value inequalities like this one are a fundamental topic in college algebra. These exercises teach valuable skills like manipulating expressions and understanding the effects of absolute value.

• Solving these problems improves your ability to tackle more advanced mathematics courses.
• Practicing college algebra regularly helps solidify foundational concepts that are crucial for success in higher-level math.

Engaging with these exercises prepares you for calculus and other math courses by developing your logical reasoning and problem-solving skills.