Problem 38 - Precalculus

Question

Rewrite each expression without using absolute value notation. $$|x+1|+4|x+3| \text { given that } x<-3$$

Step-by-Step Solution

Answer

The expression becomes \(-5x - 13\).

Step 1: Determine Absolute Value for |x + 1|

Since we know that \(x < -3\), this means \(x + 1 < -2\) because any number less than -3 plus 1 will still be less than -2. Therefore, \(|x + 1| = -(x + 1)\). This is because the absolute value of a negative number is its positive counterpart.

Step 2: Determine Absolute Value for |x + 3|

With \(x < -3\), it follows that \(x + 3 < 0\). Therefore, the absolute value \(|x + 3| = -(x + 3)\) because the absolute value turns a negative expression into its positive counterpart.

Step 3: Rewrite the Original Expression Using Steps 1 and 2

Now replace the absolute value expressions using what we determined in steps 1 and 2. The expression \(|x+1|+4|x+3|\) becomes:\[-(x + 1) + 4(-(x + 3))\]

Step 4: Simplify the Expression

Expand and simplify the expression:\[-(x + 1) + 4(-(x + 3)) = -x - 1 + 4(-x - 12)\]Simplify further:\(-x - 1 - 4x - 12 = -5x - 13\)

Key Concepts

Absolute Value Expression

An absolute value expression involves the absolute value function, which is a way to express numbers as non-negative entities. The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For example, the absolute value of both 3 and -3 is 3.

When dealing with absolute value equations, like \(|x + 1|\), one must consider the value of the expression inside absolute value brackets and compare it against zero. If the expression inside is negative, its absolute value is the positive version of that expression. If it is positive or zero, the value remains unchanged.

• For \(|x+1|\) given \(x < -3\), the expression \(x+1\) is negative. So, the absolute value \(|x+1| = -(x+1)\).
• This ensures that \(|x < -3|\) becomes a positive expression, complying with the definition of absolute value.

Negative Numbers

Negative numbers are values less than zero. They appear to the left of zero on the number line. Absolute value operations often involve negative numbers, especially when simplifying expressions.

In this exercise, we have conditions that always result in negative outcomes for the base expressions, such as \(x + 1\) and \(x + 3\).

• Since \(x < -3\), \(x + 1\) becomes a negative value because \(-3 + 1\) results in \(-2\), which is negative.
• Similarly, \(x + 3\) is also negative, since adding 3 to any number less than -3 stays negative (e.g., \(-3 + 3 = 0)\).

Recognizing whether a number or expression is negative impacts how we apply different algebraic operations, especially when simplifying expressions under absolute values.

Expression Simplification

Expression simplification involves reducing complex algebraic expressions into their simplest form. It can include substituting variables, combining like terms, and factoring.

In the given problem, expression simplification is a key step in solving and understanding the equations without absolute value symbols.

• Start by substituting the negative forms of the expressions determined by absolute values: \(-(x+1) + 4(-x-3)\).
• This is expanded to \(-x - 1 + 4(-x - 12)\).
• Combine and simplify terms to achieve the expression \(-5x - 13\).

The goal is to streamline the expression into a manageable and easily interpretable form, which helps in solving or further mathematical analysis.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (addition, subtraction, multiplication, and division). They are foundational in algebra and can represent real-world problems mathematically.

This exercise involves manipulating algebraic expressions that include variables like \(x\) within absolute values. The process of simplifying algebraic expressions involves:

• Understanding the relationships between numbers and variables.
• Applying algebraic rules, including the treatment of absolute values.
• Rewriting expressions by considering given constraints, like \(x < -3\) in this problem.

Mastering algebraic expressions allows students to translate problems into equations which are easier to analyze and solve.