1 Less Than Twice A Number.

1 Less Than Twice a Number: Exploring Mathematical Expressions and Problem-Solving

Mathematical expressions, seemingly simple at times, often hold the key to unlocking complex problems in various fields, from engineering and physics to finance and computer science. Understanding how to translate word problems into mathematical equations is a crucial skill. This article delves into the expression "1 less than twice a number," exploring its meaning, application in different scenarios, and how to solve problems involving it. We'll also discuss related concepts and provide practice examples to solidify your understanding.

Understanding the Expression: "1 Less Than Twice a Number"

The phrase "1 less than twice a number" represents a specific mathematical operation. Let's break it down step-by-step:

• A number: This refers to an unknown value, often represented by a variable, such as x, y, or n.

• Twice a number: This means multiplying the number by 2. So, if the number is x, "twice a number" is 2x.

• 1 less than: This implies subtracting 1 from the previous result.

Therefore, the complete expression "1 less than twice a number" translates mathematically to 2x - 1.

Translating Word Problems into Equations

Many real-world problems can be modeled using this expression. The key lies in carefully identifying the unknown number and the operations described in the problem statement. Let's consider some examples:

Example 1: John's age is 1 less than twice his sister's age. If his sister is 15 years old, how old is John?

Here, the unknown is John's age. Let's represent John's age by j and his sister's age by s. The problem states: j = 2s - 1. Since s = 15, we substitute this value into the equation: j = 2(15) - 1 = 30 - 1 = 29. Therefore, John is 29 years old.

Example 2: The length of a rectangle is 1 less than twice its width. If the width is 8 cm, what is the length?

Again, we can use the expression 2x - 1. Let the width be represented by w and the length by l. The equation becomes: l = 2w - 1. Substituting w = 8 cm, we get l = 2(8) - 1 = 16 - 1 = 15 cm. The length of the rectangle is 15 cm.

Solving Equations Involving "1 Less Than Twice a Number"

The expression 2x - 1 can be part of more complex equations. Let’s examine how to solve these:

Example 3: Find the value of x if 2x - 1 = 9.

To solve this equation, we need to isolate x. We can do this by performing inverse operations:

1. Add 1 to both sides: 2x - 1 + 1 = 9 + 1 => 2x = 10

2. Divide both sides by 2: 2x / 2 = 10 / 2 => x = 5

Therefore, the value of x is 5.

Example 4: Solve the equation 3(2x - 1) = 15.

In this case, we first need to distribute the 3 to the terms inside the parenthesis:

1. Distribute: 6x - 3 = 15

2. Add 3 to both sides: 6x - 3 + 3 = 15 + 3 => 6x = 18

3. Divide both sides by 6: 6x / 6 = 18 / 6 => x = 3

The solution to this equation is x = 3.

Example 5: A more complex scenario: Find x if 2x - 1 + 5x = 26.

1. Combine like terms: 7x - 1 = 26

2. Add 1 to both sides: 7x = 27

3. Divide by 7: x = 27/7

This example demonstrates that the solution might not always be a whole number.

Applications in Different Fields

The simple expression "1 less than twice a number" has surprising versatility. It finds application in various areas, including:

• Geometry: Calculating lengths, areas, and perimeters of shapes based on relationships between dimensions.

• Physics: Modeling relationships between physical quantities, such as velocity and time in uniformly accelerated motion.

• Computer Science: Creating algorithms and formulas for calculations within programs.

• Finance: Simple interest calculations, determining profit margins, and other financial modeling.

Further Exploration: Inequalities and Quadratic Equations

The expression can also be used in inequalities. For example, "1 less than twice a number is greater than 10" translates to 2x - 1 > 10. Solving this inequality involves similar steps as solving equations, but with careful consideration of inequality signs.

Furthermore, the expression could be part of a quadratic equation. For instance, (2x - 1)² = 25. Solving quadratic equations requires different techniques, often involving factoring or the quadratic formula.

Practice Problems

Here are a few practice problems to test your understanding:

1. The number of apples in a basket is 1 less than twice the number of oranges. If there are 7 oranges, how many apples are there?

2. Solve the equation: 4(2x - 1) - 6 = 10

3. Find the value of x: 2x - 1 = 2x + 5

4. Solve the inequality: 2x - 1 < 7

5. If the perimeter of a rectangle is 34 cm and the length is 1 less than twice the width, find the length and width of the rectangle.

Conclusion

The expression "1 less than twice a number," though seemingly basic, serves as a fundamental building block in mathematical problem-solving. Mastering its interpretation and application opens doors to tackling more complex problems across numerous disciplines. Consistent practice and a methodical approach to problem-solving are key to developing proficiency in this area. Remember to always carefully analyze the word problem, identify the unknown variable, translate the information into a mathematical equation, and then solve using appropriate algebraic techniques. By understanding the underlying concepts, you can confidently navigate more intricate mathematical scenarios.