Problem 68 - Elementary Algebra

Question

What numbers satisfy the condition: eight more than three times a number is less than or equal to fourteen?

Step-by-Step Solution

Answer

Answer: The range of numbers that satisfies the condition includes all numbers less than or equal to 2, including negative numbers and 2 itself.

Step 1: Identify the inequality

To translate the condition into a mathematical inequality, use the following steps: - "Eight more than" means 8+ - "Three times a number" can be represented as 3x, where x is the number in question. - "Is less than or equal to" is the inequality symbol ≤ - "Fourteen" is 14 Combining these, we get the inequality: 8 + 3x ≤ 14.

Step 2: Simplify the inequality

Now, we'll simplify the inequality by isolating x (the number we're trying to find) on one side of the inequality. 1. Subtract 8 from both sides of the inequality: 3x ≤ 6. 2. Divide both sides by 3: x ≤ 2.

Step 3: Identify the range of numbers that satisfy the condition

The inequality x ≤ 2 means any number less than or equal to 2 will satisfy the condition. So, the range of numbers includes all the numbers up to 2, including negative numbers and 2 itself.

Key Concepts

Translating Word Problems to Inequalities

When tackling word problems involving inequalities, the key is to methodically translate the given statements into a mathematical form. Let's take the sentence, 'eight more than three times a number is less than or equal to fourteen'. We start by identifying the phrases that indicate mathematical operations. For instance, whenever we see 'more than', we'll think about addition, and 'times' indicates multiplication.

The phrase 'three times a number' suggests we are working with an unknown number, often denoted by a variable like x or n. Mathematically, this would be expressed as '3x'. 'Eight more than' tacks on an additional 8, leading us to the expression '8 + 3x'. The key phrase 'is less than or equal to' tells us that this expression doesn't exceed a certain value, which translates to the ≤ inequality symbol in mathematics. Together, these pieces give us the inequality '8 + 3x ≤ 14', setting the stage for us to solve the inequality in the next steps.

Solving Linear Inequalities

Upon translating the word problem into the inequality '8 + 3x ≤ 14', solving it is the next step. To isolate the variable x, we perform operations on both sides of the inequality, akin to solving an equation, but with one crucial difference: If we multiply or divide by a negative number, we must flip the direction of the inequality sign. However, in this problem, we only deal with positive numbers and straightforward operations.

First, we subtract 8 from both sides to undo the addition, resulting in '3x ≤ 6'. Next, to determine the value of x, we divide both sides by 3, bringing us to 'x ≤ 2'. It's crucial to ensure each operation is done neatly and accurately, as a misstep could lead to the wrong inequality range. The simplicity of these operations belies the importance of understanding and following the rules of inequalities carefully.

Understanding Inequality Symbols

Inequality symbols are the backbone of problems like the one we are considering. They indicate the comparative value between two expressions or numbers. The symbol ≤ means 'less than or equal to', so in the inequality x ≤ 2, x represents a number that is either less than 2 or exactly 2.

The other common symbols include:

• < for 'less than'
• > for 'greater than'
• ≥ for 'greater than or equal to'

Understanding these symbols is crucial for interpreting and solving inequalities. For example, the range x ≤ 2 encompasses all numbers up to 2, including negative numbers and zero. It's a reminder that inequality symbols set the stage for the values included in the solution – often, a whole range rather than a single number.