Problem 22 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-2.5 x \leq-1.25\)

Step-by-Step Solution

Answer

The solution is \( (-\infty, 0.5] \).

Step 1: Isolate the Variable

To solve the inequality \(-2.5x \leq -1.25\), first isolate the variable \(x\) by dividing both sides of the inequality by \(-2.5\).

Step 2: Perform the Division

Divide both sides of the inequality by \(-2.5\). Remember to flip the inequality symbol because you are dividing by a negative number: \(\frac{-1.25}{-2.5} \geq x\).

Step 3: Simplify the Expression

Simplify the right-hand side: \(0.5 \geq x\). This can be written as: \(x \leq 0.5\).

Step 4: Express the Solution in Interval Notation

The solution \(x \leq 0.5\) in interval notation is \((-\infty, 0.5]\).

Step 5: Graph the Solution Set

Graph the interval \((-\infty, 0.5]\) on the number line. Draw a solid dot at \(0.5\) and shade all the numbers to the left of \(0.5\).

Key Concepts

Inequality Notation

Inequalities are similar to equations but instead of an equal sign, they use symbols like <, >, ≤, or ≥. In the exercise \(-2.5x ≤ -1.25\), our goal is to find the values of \(x\) that make the inequality true. When you see \(<\), it means 'less than.' When you see ≠, it means 'less than.' When you see \(≤\), it means 'less than or equal to.' And when you see \(≥\), it means 'greater than or equal to.' These notations help us express ranges of numbers that meet certain conditions. Always pay special attention to the direction of the inequality sign as it indicates the relationship between the two quantities.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It provides a concise way to describe intervals. For example, in the solution \((-∞, 0.5]\), the interval starts from negative infinity and goes up to and includes 0.5. Here's how to read interval notation:

• ( ) or \( \): These mean the endpoint is not included.
• [ ] or \( \): These mean the endpoint is included.
• ∞ or \( \): Represents infinity or negative infinity, which are always written with parentheses, because infinity is not a specific, attainable number.

This notation helps streamline the representation of solutions derived from inequalities.

Graphing Inequalities

Graphing inequalities visually shows the solution set on a number line. In our solution, we graph \((-∞, 0.5]\). To do this:

• First, draw a solid dot at 0.5 to indicate it is included in the solution.
• Next, shade the line to the left of 0.5, extending towards negative infinity.

The solid dot tells us 0.5 is part of the solution (denoted by \[ \]), while the shaded part indicates all numbers less than 0.5. Graphing helps visually understand the range of numbers that satisfy the inequality.

Flipping Inequality Sign

When solving inequalities, one key concept is the rule for flipping the inequality sign. If you multiply or divide both sides of the inequality by a negative number, you must flip the direction of the inequality sign. In the given exercise, we divided by \(-2.5\), so:

• Original: \(-2.5x ≤ -1.25\)
• Divide both sides by \(-2.5\): \( x ≥ 0.5\)

This step ensures the inequality relationship remains true. It's crucial because failing to flip the sign changes the meaning of the inequality, leading to incorrect solutions. Always remember: when multiplying or dividing by negative numbers, flip the inequality sign!