Problem 37 - Precalculus

Question

Rewrite each expression without using absolute value notation. $$|x+1|+4|x+3| \text { given that }-\frac{5}{2}

Step-by-Step Solution

Answer

The rewritten expression is \(3x + 11\).

Step 1: Analyze the Range of x

We are given the range \(-\frac{5}{2} < x < -\frac{3}{2}\). This means \(x\) is between \(-2.5\) and \(-1.5\). We need to check where \(x + 1\) and \(x + 3\) fall in this range.

Step 2: Determine Sign of Each Expression

For \(x + 1\): Since \(x\) in \(-\frac{5}{2} < x < -\frac{3}{2}\), adding 1 gives \(-\frac{5}{2} + 1 < x + 1 < -\frac{3}{2} + 1\) which simplifies to \(-\frac{3}{2} < x + 1 < -\frac{1}{2}\). Hence, \(x+1 < 0\).For \(x + 3\): Adding 3 gives \(-\frac{5}{2} + 3 < x + 3 < -\frac{3}{2} + 3\) which simplifies to \(\frac{1}{2} < x + 3 < \frac{3}{2}\). Hence, \(x+3 > 0\).

Step 3: Rewrite the Absolute Values

Because \(x+1 < 0\), \(|x+1| = -(x+1)\).Because \(x+3 > 0\), \(|x+3| = x+3\).

Step 4: Substitute and Simplify the Expression

Substitute the absolute value expressions:\[ |x+1| + 4|x+3| = -(x+1) + 4(x+3) \]Simplify the expression:\[ = -x - 1 + 4x + 12 \]Combine like terms:\[ = 3x + 11 \]

Step 5: Final Step: Verify and Conclude

Check the expressions based on the given condition. Since we determined the signs correctly for the given range of \(x\), the rewritten expression without absolute values is \(3x + 11\).

Key Concepts

Understanding Inequality in Expressions

Inequality statements describe mathematical expressions in which one side is not necessarily equal to the other, but instead may be greater or lesser. In the initial exercise, we encountered an inequality: \(-\frac{5}{2} < x < -\frac{3}{2}\). This inequality tells us that our variable, \(x\), can be any number between \(-2.5\) and \(-1.5\). In practical terms, inequalities help define a range of possible values, rather than a single solution.
They often appear in problems that involve absolute values, as they allow us to understand the behavior of expressions within specific limits.

• If you're solving an inequality involving absolute values, always evaluate how the expressions within the absolute value will behave over the specified interval.
• Consider changes in sign, as this affects whether terms in absolute values need to be negated or left as is.

In this exercise, determining whether \(x+1\) or \(x+3\) are positive or negative guides us in rewriting absolute values correctly.

Algebraic Manipulation of Expressions

Algebra plays a crucial role when transforming expressions, especially when shifting between forms, like from absolute values to simpler expressions. It requires us to apply rules to manage terms correctly.
In our exercise, we worked through several steps. One key task was to rewrite expressions like \(|x+1|\) and \(|x+3|\) based on whether their inner values were positive or negative within the interval given by the inequality.

• If \(x+1\) is negative, like in our exercise, the absolute value \(|x+1|\) becomes \(-(x+1)\).
• If \(x+3\) is positive, \(|x+3|\) remains unchanged as \(x+3\).

Once rewritten, algebraic simplification involves collecting and combining like terms neatly. Here, after simplification, we've arrived at the expression \(3x + 11\). Understanding each step and ensuring the confident application of rules is essential for solving algebraic problems effectively.

Decoding Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that signify a particular quantity or a relationship. They are the building blocks of more complex algebra and reasoning, allowing for detailed description of mathematical scenarios.
In our task, the given expression with absolute values was \(|x+1|+4|x+3|\). Understanding it involves recognizing the relationships and operations involved, like addition, multiplication, and the absolute value operation.

• Decoding expressions means determining how different parts, such as \(x+1\) and \(x+3\), influence the whole based on \'x\'s potential values.
• Each part of the expression carries a function - here, \(x+1\) and \(x+3\) indicate how values shift due to \(x\).

Simplifying the mathematical expression allows for clearer understanding and application. In the example, recognizing the absolute values and transforming them helped convert a potentially complex expression into a simple linear form \(3x + 11\), facilitating more accessible insights into the problem.