Problem 52 - Elementary Algebra Within Reach

Question

Translate the verbal statement into a linear inequality. \(x\) is at least \(-3\) and at most 5 .

Step-by-Step Solution

Answer

The linear inequality that represents the verbal statement is \(-3 \leq x \leq 5\).

Step 1: Translating 'at least'

The statement 'x is at least -3' means that x is greater than or equal to -3. We write it as \(x \geq -3\).

Step 2: Translating 'at most'

The statement 'x is at most 5' means that x is less than or equal to 5. We write it as \(x \leq 5\).

Step 3: Combining the two inequalities

Since x must satisfy both inequalities, we can write them combined as \(-3 \leq x \leq 5\). As long as x is any number in between -3 and 5 (including -3 and 5), it will satisfy both inequalities.

Key Concepts

Verbal Statement Translation

Verbal statements describe situations in words, which we often need to convert into mathematical expressions to solve problems. This process is known as verbal statement translation. Understanding the meaning behind words like "at least" and "at most" is key in this translation. When we say that " x is at least -3 ", it means that x can be any number greater than or equal to -3. Similarly, " x is at most 5 " suggests that x can be any number less than or equal to 5.

• "At least" means greater than or equal to.
• "At most" means less than or equal to.

Converting these verbal phrases into mathematical terms allows us to better analyze and understand the problem at hand.

Inequality Symbolism

To translate verbal statements into mathematical expressions, we use inequality symbols. These symbols help us describe a range of values rather than specific numbers. In the context of inequalities, there are a few symbols you will often encounter:

• ">" indicates 'greater than'
• "<" indicates 'less than'
• "\geq" stands for 'greater than or equal to'
• "\leq" stands for 'less than or equal to'

For the statement " x is at least -3 ", we use the symbol \geq to write \( x \geq -3 \).
For " x is at most 5 ", the symbol is \leq , leading to \( x \leq 5 \).These symbols create a framework that allows us to succinctly communicate the conditions that a variable like x must satisfy.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and symbols that represent mathematical relationships. In our example, the expressions \( x \geq -3 \) and \( x \leq 5 \) are both algebraic expressions. They utilize variables (in this case, x) to denote values that meet specific conditions. There are numerous ways we can manipulate these expressions depending on the problem requirements. In this particular context:

• \( x \geq -3 \) tells us that any value of x must be no smaller than -3.
• \( x \leq 5 \) instructs that any value of x must not exceed 5.

Each part conveys a condition for x, giving it the flexibility to range between these limits.

Inequality Solution Steps

Solving inequalities involves several steps, primarily geared towards finding the values of the variable that meet given conditions. Let's break down the steps for our specific problem:**Step 1: Translate the Verbal Statement**
We start by transforming the verbal expressions "at least -3" and "at most 5" into their corresponding mathematical inequalities, \( x \geq -3 \) and \( x \leq 5 \).**Step 2: Combine the Inequalities**
Since x needs to satisfy both conditions, we write a compound inequality, \( -3 \leq x \leq 5 \). This expression succinctly wraps both conditions into one, showing that x should lie somewhere between -3 and 5, inclusive.These steps provide a simple roadmap for translating a verbal condition into a solvable mathematical inequality, allowing us to define the range of possible solutions for x.