Problem 50 - Algebra 2

Question

Determine whether each statement is sometimes, always, or never true. Explain your reasoning. If \(a, b,\) and \(c\) are real numbers, then \(c|a+b|=|c a+c b|\)

Step-by-Step Solution

Answer

Sometimes true; it holds when \( c \geq 0 \).

Step 1: Understand The Equation

The given equation is \( c|a+b| = |c a + c b| \). We are asked to determine if this equation is sometimes, always, or never true for any real numbers \( a, b, \text{ and } c \).

Step 2: Simplify The Right Side

Notice that the right side of the equation, \( |c a + c b| \), can be rewritten using the distributive property of addition which gives us \( |c(a+b)| \).

Step 3: Use The Absolute Value Property

Recall the property of absolute value: \( |kz| = |k||z| \) where \( k \) and \( z \) are real numbers. Applying this to the right side, we have \( |c(a+b)| = |c||a+b| \).

Step 4: Compare Both Sides

Now our equation is \( c|a+b| = |c||a+b| \). For both sides to be equal, \( c \) must satisfy \( |c| = c \).

Step 5: Determine The Condition For Equality

The equation \( |c| = c \) is true when \( c \geq 0 \) because the absolute value of a number is equal to the number itself when it is non-negative.

Step 6: Conclusion

Thus, the equation \( c|a+b| = |c a + c b| \) is sometimes true. It is true only when \( c \geq 0 \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, stating that multiplication distributes over addition or subtraction. This means if you multiply a sum by a number, it's the same as multiplying each addend by the number and then adding the products. For example, the distributive property is expressed as:

• \( a(b + c) = ab + ac \).

In the original exercise, the property comes into play when rewriting the right side of the equation using distributive property. We changed \(|ca + cb|\) to \(|c(a+b)|\). This simplification step is crucial for understanding how both sides of the equation relate when considering absolute values. By recognizing and applying the distributive property, we transform seemingly complex expressions into more manageable forms.

Real Numbers

Real numbers are a set of numbers that include all the rational numbers, such as positive and negative integers, fractions, and all irrational numbers, such as the square roots and the number \( \pi \). They form a continuous number line, extending infinitely in both the positive and negative directions.

• Real numbers are capable of representing any value on the continuous number line.
• They include numbers like 0, 1, -5, 1.25, and \(\sqrt{2}\).

In the context of our exercise, \(a\), \(b\), and \(c\) are specified as real numbers. This means any conclusion derived from the equation holds for this entire set. The property explored in the equation relies on the characteristics of real numbers, especially when it comes to utilizing absolute values and properties such as the distributive rule.

Equation Simplification

Simplifying equations is a process in which we break down complex algebraic expressions into simpler parts. This makes equations easier to solve or analyze. Simplification often involves applying algebraic rules like the distributive property, combining like terms, and factoring. In the exercise, equation simplification took place when rewriting \(|ca + cb|\) as \(|c(a+b)|\). By applying the absolute value property \(|kx| = |k||x|\), the equation \(c|a+b| = |c(a+b)|\) simplified further to \(c|a+b| = |c||a+b|\). Our focus was then on the condition \(|c| = c\), leading us to conclude that the equation is sometimes true, holding true specifically when \(c \geq 0\).

• Always aim to express equations in their simplest form.
• Simplification helps in understanding the underlying relationships and solutions.