Problem 2 - Algebra 2 and Trigonometry

Question

Explain why the solution set of \(|2 x+4|+7<3\) is the empty set.

Step-by-Step Solution

Answer

The inequality requires an impossible condition of negative absolute value, leading to an empty solution set.

Step 1: Understand Absolute Value Inequality

The absolute value expression \(|2x+4|\) represents the distance of \(2x+4\) from zero on the number line. Therefore, any expression like \(|2x+4|\) being equal to or greater than zero, because distance is always non-negative.

Step 2: Isolate the Absolute Value Expression

Start by subtracting 7 from both sides of the inequality \(|2x+4| + 7 < 3\). This results in: \(|2x+4| < -4\).

Step 3: Analyze the Simplified Inequality

The inequality \(|2x+4| < -4\) asks for the absolute value to be negative, but since absolute values are always non-negative, there are no real numbers that satisfy this inequality.

Step 4: Conclusion About the Solution Set

Since no real number \(x\) can make \(|2x+4|\) negative, it implies that the original inequality \(|2x+4| + 7 < 3\) has no solution. The solution set is therefore empty.

Key Concepts

Solution Sets

In mathematics, a solution set refers to the collection of all possible values that satisfy an equation or inequality. In the case of absolute value inequalities, the solution set contains all the real numbers that make the inequality true. For instance, when solving the inequality |2x+4|+7<3, we seek values of x that meet this condition.

• However, as seen in the provided solution, we find that no real numbers can satisfy the inequality.
• This is because the modified inequality |2x+4| < -4 is fundamentally impossible due to the properties of absolute values.

In such cases where no possible values fulfill the condition, the solution set is deemed empty. An empty solution set signifies that there are no solutions that meet the original inequality.

Real Numbers

Real numbers form a vast and continuous set of numbers used to represent quantities in mathematics. They include all the numbers we commonly use, such as integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\).

• Real numbers can be both positive and negative, but importantly, they also include zero.
• In solving absolute value inequalities, we seek real numbers that satisfy given mathematical conditions.

In the case of the inequality |2x+4| < -4, real numbers are employed to explore potential solutions. However, since it's impossible for any real number to produce a negative absolute value, no solutions exist within this set.

Inequalities

An inequality is a mathematical statement that shows the relationship between two expressions by using signs like \(<\), \(>\), \(\leq\), or \(\geq\). Inequalities help us understand bounds or ranges of solutions.

• When solving an inequality, we determine which values satisfy the given condition.
• With absolute value inequalities, the solutions are entirely dependent on understanding the properties of absolute values, as these significantly impact the inequality's solvability.

For the inequality |2x+4|+7<3, solving it required isolating the absolute value and analyzing the resulting inequality. In this case, the expression |2x+4| < -4 demonstrates an unachievable condition, highlighting the absence of a solution.

Absolute Value Properties

Absolute value describes the distance of a number from zero, and is always non-negative. It is depicted as |a|, where \(a\) could be any number, positive or negative. Key properties include:

• Every absolute value is greater than or equal to zero.
• Absolute values measure the magnitude without considering direction.

When dealing with absolute value inequalities, these properties are crucial. The exercise |2x+4| < -4 requires us to find a real number whose distance from zero is negative, which is impossible due to the non-negative nature of absolute values. Thus, it demonstrates how understanding these properties can lead to determining whether solutions are feasible or not.