Problem 12 - Algebra A Combined Function

Question

Graph each inequality on the number line. $$ -4

Step-by-Step Solution

Answer

Graph \(-4\) with an open circle, \(0\) with a closed circle, and shade between them.

Step 1: Understand the Inequality

The given inequality is \(-4 < x \leq 0\). It represents the set of all numbers \(x\) that are greater than \(-4\) but less than or equal to \(0\). We need to find a way to graph this set of numbers on a number line.

Step 2: Identify the Endpoints of the Interval

The endpoints of the interval are \(-4\) and \(0\). The symbol \('<'\) indicates that \(x\) is not equal to \(-4\), so the interval does not include \(-4\). The symbol '\(\leq\)' indicates that \(x\) is equal to or less than \(0\), including \(0\) in our interval.

Step 3: Plot the Interval on the Number Line

Plot a number line and mark the numbers \(-4\) and \(0\). Since \(x\) is greater than \(-4\) but not including \(-4\), use an open circle at \(-4\). Since \(x\) is less than or equal to \(0\), use a closed or filled circle at \(0\).

Step 4: Shade the Region Between the Endpoints

Shade the region between \(-4\) and \(0\) on the number line to represent all values of \(x\) that satisfy the inequality. The shading shows the range of numbers that \(x\) can take, starting just after \(-4\) and including \(0\).

Key Concepts

Number Line

The number line is a visual representation that helps us to understand the position of numbers relative to each other. It's like a ruler laid out in a straight line, with numbers increasing as you move to the right and decreasing as you move to the left. This simplicity makes it an excellent tool for visualizing inequalities.When dealing with inequalities, we plot specific points and shade regions on the number line to represent solutions. For example, with the inequality \(-4 < x \leq 0\), the number line helps us see that \(x\) can be any value between \(-4\) and \(0\), but not including \(-4\). This clarity is crucial, especially when learning to understand more complex inequalities.

Graphing Inequalities

Graphing inequalities involves visually displaying the range of values that satisfy an inequality on a number line. It requires understanding how different parts of the inequality correspond to actions on the number line.

• Start by locating the endpoints of your interval. For \(-4 < x \leq 0\), these are \(-4\) and \(0\).
• Use an open circle to indicate that a number is not included in the set. At \(-4\), place an open circle, because \(x\) can never be \(-4\) itself.
• Use a closed circle to show that a number is part of the solution. At \(0\), place a closed circle, because \(x\) can be \(0\) according to \(x \leq 0\).

After marking the endpoints, shade the area on the number line between them to represent all possible values for \(x\). The shaded part is what makes the inequality visible and comprehensible, bridging numbers with actual solutions.

Interval Notation

Interval notation provides a succinct way to represent ranges of values, particularly when dealing with inequalities like \(-4 < x \leq 0\). It's a compact alternative to describing sets verbally or graphically.An interval consists of two numbers within brackets. These numbers are the endpoints, and the type of brackets indicates whether the endpoints are included or excluded:

• \((\) or \()\): used for open endpoints, meaning the endpoint is not included. For example, in \(-4 < x\), \(-4\) is not part of the solution.
• \([\) or \()]\): used for closed endpoints, where the endpoint is included. Since \(x \leq 0\), \(0\) is part of the solution, hence a closed bracket is used.

Putting this together, the interval notation for the inequality \(-4 < x \leq 0\) is \((-4, 0]\). It clearly expresses the set of numbers without needing additional words, ideal for solving and communicating math problems concisely.