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Chapter 2: Problem 47

Solve. Write the solution set using interval notation. See Examples 1 through 7. $$ 0.8 x+0.6 x \geq 4.2 $$

Short Answer

Expert verified
The solution set in interval notation is \([3, \infty)\).

Step by step solution

01

Combine Like Terms

The given inequality is \(0.8x + 0.6x \geq 4.2\). To simplify, combine the terms with \(x\): \((0.8 + 0.6)x \geq 4.2\). This simplifies to \(1.4x \geq 4.2\).
02

Isolate the Variable

To solve for \(x\), divide both sides of the inequality \(1.4x \geq 4.2\) by 1.4. This gives us \(x \geq \frac{4.2}{1.4}\).
03

Perform the Division

Calculate \(\frac{4.2}{1.4}\). Dividing 4.2 by 1.4 results in \(x \geq 3.0\).
04

Write the Solution in Interval Notation

The inequality \(x \geq 3\) means \(x\) includes all numbers greater than or equal to 3. In interval notation, this is written as \([3, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a mathematical way of writing down intervals on the number line. It is commonly used to specify the solution set of inequalities in a concise form. Let's break down how it works:

Consider the inequality expression below:
  • For an inequality like \(x \geq 3\), we use a bracket \([3, \infty)\) to show that \(x\) can equal 3 and any number larger than 3. This is called a closed bracket because it includes the endpoint.
  • If the inequality was \(x > 3\), the notation would be \((3, \infty)\) with parenthesis to show that 3 is not included, known as an open bracket.
  • Infinity \((\infty)\) is always paired with a parenthesis, as infinity itself is not a number but a concept of boundlessness.
Interval notation helps make solutions neat and easy to read, especially when working with complex inequalities.
Combining Like Terms
Combining like terms is a crucial step for simplifying algebraic expressions and equations. It involves merging terms that have the same variable raised to the same power. This simplification is important for solving equations systematically. Here's how it's applied:

In the example \(0.8x + 0.6x \geq 4.2\):
  • Both \(0.8x\) and \(0.6x\) are "like terms" because they each include \(x\) raised to the power of 1.
  • Combine them by adding their coefficients: \(0.8 + 0.6 = 1.4\). Hence, \(0.8x + 0.6x\) becomes \(1.4x\).
This process of adding the coefficients simplifies the problem and makes further steps more straightforward. It's the foundational step for solving not only equations but inequalities too.
Solving Inequalities
Solving inequalities is somewhat similar to solving equations, but the direction of the inequality symbol must be carefully considered. Here鈥檚 a step-by-step snapshot:
  • Start with combining like terms if needed, as seen with \(1.4x \geq 4.2\).
  • To isolate the variable \(x\), perform arithmetic operations such as addition, subtraction, multiplication, or division. In our example, dividing both sides by 1.4 simplifies it to \(x \geq 3\).
  • Remember that dividing or multiplying both sides of an inequality by a negative number would reverse the inequality sign, but that is not a concern here.
  • Finally, you express the solution in interval notation as \([3, \infty)\), capturing all numbers \(x\) that satisfy the inequality.
The process necessitates attention to the inequality sign, ensuring accuracy in representing the range of solutions. Thus, it's essential to approach each step methodically.

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