Problem 24 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2. $$ 4 x<-36 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -9)\).

Step 1: Isolate the Variable

In order to solve the inequality, we want to isolate the variable, which is \(x\). The original inequality is \(4x < -36\). We need to divide both sides of the inequality by \(4\) to isolate \(x\).

Step 2: Divide Both Sides by 4

By dividing both sides of the inequality \(4x < -36\) by \(4\), we get: \[ x < \frac{-36}{4} \] Simplifying the right side gives \(x < -9\).

Step 3: Represent the Solution Set on a Number Line

The solution \(x < -9\) means that \(x\) can be any number less than \(-9\). On a number line, you would indicate this by drawing an open circle at \(-9\) (indicating that \(-9\) is not included in the solution set) and shading the line to the left of \(-9\).

Step 4: Write the Solution in Interval Notation

The interval notation for \(x < -9\) is \((-\infty, -9)\). This notation represents all numbers less than \(-9\).

Key Concepts

Interval Notation

Interval notation is a concise way to represent a range of values that satisfy an inequality. Instead of listing all possible values, interval notation uses certain symbols to define the bounds of the range. Here is how it works:

• Parentheses \((\) or \()\) indicate that a number is not included in the set. For example, in the solution \((-infty, -9)\), this indicates all numbers less than -9, but not including -9 itself.
• Brackets \([\) or \()]\) would be used if the endpoint is included in the solution set, marked as a closed interval.

Interval notation makes it easy to understand the range at a glance. It is often used because it is both compact and clear, particularly when describing solutions to linear inequalities such as \( x < -9 \). In this example, \((-infty, -9)\) provides a clear representation of all numbers satisfying the inequality.

Graphing Inequalities

Graphing inequalities involves visually representing the range of values that satisfy an inequality on a coordinate plane or number line. To correctly graph an inequality, especially a simple one-variable inequality like \(x < -9\), follow these steps:

• Identify the boundary number: Here, it is -9.
• Decide whether to use an open circle or a closed dot on the number line:
  • Use an open circle for inequalities like \(<\) or \(>\), showing the boundary number is not included in the solution.
  • Use a closed dot for \(\leq\) or \(\geq\), indicating inclusion.
• Shade the appropriate side of the number line:
  • For \(<\), shade to the left.
  • For \(>\), shade to the right.

This visual representation helps in easily identifying all potential solutions of the inequality at a glance.

Number Line Representation

A number line is a simple yet powerful tool to visually express inequalities. When representing \(x < -9\) on a number line, here's how you can do it:
Start by drawing a horizontal line and marking even, equally spaced increments. Locate the position of -9 among these increments.

• Draw an open circle on -9 to indicate that this point is not included in the solution set.
• Shade the line extending from -9 to the left towards negative infinity. This shaded area visually indicates all numbers less than -9 fulfilling the inequality.

Using a number line for inequalities not only provides a clear picture of the solution but also ensures comprehensive understanding by visually setting apart included and excluded values. It is an intuitive way for students to connect algebraic inequalities with visual data, thereby reinforcing understanding.