Problem 21 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-1.3 x \geq-5.2\)

Step-by-Step Solution

Answer

The solution is x \leq 4, graphed as (-\infty, 4].

Step 1 - Isolate the variable

To isolate the variable on one side of the inequality, divide both sides of the inequality by -1.3. Be aware that dividing by a negative number reverses the inequality sign:\[ \frac{-1.3x}{-1.3} \leq \frac{-5.2}{-1.3} \]This simplifies to:\[ x \leq 4 \]

Step 2 - Graph the solution set

Draw a number line and identify the point corresponding to x = 4. Since the inequality is \( x \leq 4 \), shade the entire region to the left of 4, and include 4 by using a closed circle at x = 4.

Step 3 - Write the solution in interval notation

Since the solution includes all values less than or equal to 4, the interval notation for the solution set is: \( (-\infty, 4] \).

Key Concepts

inequality solutions

Inequalities are similar to equations, but they do not have a single solution. Instead, they represent a range of possible values. To solve an inequality, the goal is to isolate the variable on one side.
Inequalities use symbols like \(<, >, \leq, \text{and} \geq\) to compare values.

• In the example \(-1.3x \geq -5.2\), we start by isolating \(x\).
• Remember that when you multiply or divide both sides by a negative number, you must reverse the inequality sign.
• For this problem, we divided by -1.3 resulting in \(x \leq 4\).
• This means \(x\) can be any number less than or equal to 4.

Understanding these basic steps will help you tackle more complex inequalities.

graphing inequalities

Graphing inequalities helps visualize their solutions on a number line or coordinate plane. In our example, we solved for \(x \leq 4\). To graph this:

• Draw a number line with points marked at regular intervals.
• Identify the point corresponding to 4.
• Since \(x \leq 4\), place a closed circle at 4 to indicate it is included in the solution.
• Shade the area to the left of 4, representing all numbers less than 4.

This visual representation makes it clear which values satisfy the inequality.

interval notation

Interval notation is a shorthand way to write the solution set of an inequality. It uses parentheses and brackets to describe intervals:

• Parentheses \(( \text{ or } )\) exclude an endpoint, meaning the endpoint itself is not part of the interval.
• Brackets \([ \text{ or } ]\) include an endpoint, meaning the endpoint itself is part of the interval.

For our inequality \(x \leq 4\):

• The solution set includes all numbers less than or equal to 4.
• In interval notation, this is written as \((-\text{inf}, 4]\).
• The \([\) means the set includes 4, and \((-\text{inf}\) represents all numbers extending to negative infinity.

Interval notation is a very efficient way to express solution sets clearly and concisely.