Problem 31 - Algebra 2

Question

Define a variable and write an inequality for each problem. Then solve. The difference of one half of a number and 7 is greater than or equal to 5

Step-by-Step Solution

Answer

\( x \geq 24 \)

Step 1: Define the Variable

Let the variable \( x \) represent the unknown number that we need to find.

Step 2: Write the Expression

Express one half of the number using the variable \( x \). The expression is \( \frac{1}{2}x \).

Step 3: Formulate the Inequality

The problem states that the difference of one half of the number and 7 is greater than or equal to 5. Therefore, the inequality is \( \frac{1}{2}x - 7 \geq 5 \).

Step 4: Solve the Inequality

First, add 7 to both sides of the inequality: \( \frac{1}{2}x - 7 + 7 \geq 5 + 7 \) simplifies to \( \frac{1}{2}x \geq 12 \).

Step 5: Eliminate the Fraction

Multiply both sides of the inequality by 2 to solve for \( x \): \( 2 \times \frac{1}{2}x \geq 12 \times 2 \), which simplifies to \( x \geq 24 \).

Key Concepts

Defining Variables

Defining a variable is a fundamental step when tackling any math problem that involves an unknown quantity. In our problem, we're tasked with finding an unknown number, so we introduce a variable to represent it. Here, we use the letter \( x \) to stand for this unknown number. By defining \( x \), we create a foundation that allows us to write mathematical expressions and equations to describe what we're trying to find.
In general, choosing a variable is all about setting a label for the unknown value.
Most commonly, letters like \( x \), \( y \), or \( z \) are used. Remember, the variable is just a placeholder that helps you work through the problem and will eventually lead you to an actual number.

Writing Expressions

Writing expressions involves taking the information given in a problem and translating it into mathematical language. Here, we know that one half of our unknown number \( x \) is part of the problem. This is expressed mathematically as \( \frac{1}{2}x \).
Mathematical expressions are versatile and can involve variables, numbers, and operations such as addition, subtraction, multiplication, or division. They serve as building blocks for more complex equations and inequalities. In any given problem, identifying the operations and relationships between numbers and variables helps write accurate expressions, which paves the way for solving the problem.

Formulating Inequalities

Formulating inequalities allows us to compare expressions and determine the range of possible solutions. In this exercise, we're given a situation where one half of a number minus 7 is greater than or equal to 5. The task is to set up this scenario as an inequality to find the possible values of \( x \).
This specific wording translates to the inequality \( \frac{1}{2}x - 7 \geq 5 \). This inequality expresses that the difference calculated must be at least 5 or greater. When formulating inequalities, carefully translate verbal statements by identifying keywords like 'greater than', 'less than', 'at least', or 'at most', and use corresponding mathematical signs such as \( >, <, \geq, \leq \). This precise formulation helps to accurately solve any inequalities for the given math problem.

Solving Inequalities with Fractions

Solving inequalities that include fractions involves a couple of extra steps compared to solving simple linear equations. It's important to isolate the variable on one side while balancing both sides of the inequality.
In our problem, after formulating the inequality as \( \frac{1}{2}x - 7 \geq 5 \), the first step is to eliminate constants on the side with the variable. Here, this means adding 7 to both sides to simplify the inequality to \( \frac{1}{2}x \geq 12 \).
Next, you need to deal with the fraction. Multiply each side of the inequality by the reciprocal of the fraction's coefficient, in this case, multiply by 2. This simplifies the inequality to \( x \geq 24 \), giving the solution. Remember, while solving inequalities, any multiplication or division by a negative number requires flipping the inequality sign. However, since we're dealing with positive numbers throughout, the sign remains the same.