Problem 23 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(5 x+2 \leq-48\)

Step-by-Step Solution

Answer

The solution is \(x \leq -10\) and the interval notation is \((-\infty, -10]\).

Step 1 - Subtract 2 from both sides

To isolate the term with the variable, first subtract 2 from both sides of the inequality: \[5x + 2 - 2 \ \leq -48 - 2\]

Step 2 - Simplify both sides

Combine like terms to simplify the inequality: \[5x \ \leq -50\]

Step 3 - Divide by 5

To solve for \(x\), divide both sides by 5: \[x \ \leq \frac{-50}{5}\]

Step 4 - Simplify

Simplify the division: \[x \ \leq -10\]

Step 5 - Graph the solution

On a number line, shade the region to the left of \(-10\), including the point \(-10\) itself, because the inequality is \(\leq\).

Step 6 - Write the solution in interval notation

Since \(x\) can be any value less than or equal to \(-10\), the solution in interval notation is: \[(-\infty, -10]\]

Key Concepts

inequalities

An inequality is a mathematical expression showing that two values are not equal, and they include symbols like <, >, ≤, or ≥.
In this exercise, we are given the inequality: 5x + 2 ≤ -48.
Solving an inequality is similar to solving an equation, with some important differences. Always be careful when multiplying or dividing both sides by a negative number, as this reverses the inequality sign.

interval notation

Interval notation is a concise way of expressing a range of values. The range is written in parentheses or brackets depending on whether endpoints are included or not.
For example:

• (a, b): a and b not included
• [a, b]: both a and b included
• (a, b]: a excluded, b included
• [a, b): a included, b excluded

In our solution, we found that x ≤ -10.
So, the interval notation is (-∞, -10] where -10 is included (shown with a bracket), and ∞ is always preceded by a parenthesis as it represents boundlessness.

graphing solutions

Graphing solutions of inequalities helps to visually understand the range of possible answers.
For the inequality x ≤ -10:

• Draw a number line
• Locate the point -10 and put a closed circle on it (since ≤ includes -10)
• Shade the region to the left of -10, extending to -∞ to show all numbers less than or equal to -10

This way, anyone looking at the graph can immediately understand the solution set without solving the inequality again.