Problem 3 - Elementary Algebra

Question

Graph the inequality \(-3 m+1<13\).

Step-by-Step Solution

Answer

Answer: The solution of the inequality is m > -4. On the graph, it is represented by an open circle at -4 on the number line, with shading to the right.

Step 1: Solve the inequality for "m"

To solve the inequality for "m", we want to isolate the "m" term on one side of the inequality. We do this by first subtracting 1 from both sides of the inequality: \(-3m+1-1<13-1\) \(-3m<12\) Now, we divide both sides of the inequality by -3 to solve for "m". When we divide by a negative number, we must remember to reverse the inequality sign. \(\frac{-3m}{-3}>\frac{12}{-3}\) \(m>-4\)

Step 2: Determine the boundary condition

In this inequality, the symbol is ">", so the solution does not include the value "m = -4". This indicates an open circle at the boundary point on the graph.

Step 3: Graph the inequality on a coordinate plane

We can now graph the inequality on a number line. 1. Draw a number line, and label it with numbers from at least -6 to 2. 2. Locate -4 on the number line. Since the inequality is strictly greater than -4 and not inclusive, we place an open circle at -4. 3. Shade the region to the right of -4 on the number line, as it represents the values of "m" that are greater than -4. The graph of the inequality \(-3m+1<13\) has an open circle at -4 and is shaded to the right, representing the solution \(m>-4\).

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations but with a few additional rules. In equations, the goal is to isolate the variable on one side to find its precise value. Comparatively, inequalities allow for a range of values that make the inequality true.

Let's take a closer look at the inequality \(-3m+1<13\). The first step is to isolate the variable (here, "m"). We do this by performing algebraic operations:

• Subtract 1 from both sides to get \(-3m < 12\).
• Divide both sides by -3 to solve for "m", which gives us \(m > -4\).

When dividing by a negative number, a critical rule is that the inequality sign must flip. So, \(<-\) becomes \(>-\).

Remember these steps whenever you attempt to isolate the variable while solving inequalities.

Number Line

A number line is a visual representation of numbers in a straight, horizontal line where each point corresponds to a number. It is a powerful tool for understanding inequalities.

When graphing the solution to an inequality like \(m > -4\), the number line helps us see which values "m" can take. Here's how you graph it on a number line:

• Draw a horizontal line and evenly space the numbers along it.
• Make a clear mark at the boundary point, which is -4 in this case.
• Since the condition is strictly greater than (" > "), place an open circle at -4.
• Shade the area to the right of -4 to denote that all these points satisfy the inequality.

This visualization helps understand that any number greater than -4 is a solution. It shows both the boundary and the direction in which the inequality holds true.

Boundary Conditions

Boundary conditions in inequalities define the limits of the solution set. They help determine whether the boundary is included in the solution.

In our example, the inequality \(m > -4\) has a boundary at -4. Since the inequality is strict (using the greater than \(>\) sign rather than \(\geq\)), the boundary value is not included in the solution set, hence the open circle on the number line.

When representing boundary conditions visually:

• An open circle indicates that the boundary point itself is not part of the solution. Here, it reflects \(m > -4\).
• A closed or filled circle would indicate that the boundary point is included (such as in \(m \geq -4\)).

This distinction is essential for correctly interpreting and graphing inequalities, as it precisely indicates whether the boundary is part of the values that satisfy the inequality.