Problem 45 - Algebra A Combined Function

Question

Determine whether (1,1) is included in each graph. $$ y \geq-\frac{1}{2} x $$

Step-by-Step Solution

Answer

Yes, (1,1) is included in the graph of the inequality.

Step 1: Identify the Inequality Condition

The given inequality is \( y \geq -\frac{1}{2}x \). This describes a region in the coordinate plane that includes all points where the y-coordinate is greater than or equal to \(-\frac{1}{2}\) times the x-coordinate value.

Step 2: Substitute the Point into the Inequality

We need to check whether the point \((1, 1)\) satisfies the inequality. Substitute \(x = 1\) and \(y = 1\) into the inequality: \(1 \geq -\frac{1}{2}(1)\).

Step 3: Simplify the Expression

Simplify the right side of the inequality: \(-\frac{1}{2}(1) = -\frac{1}{2}\). The inequality becomes \(1 \geq -\frac{1}{2}\).

Step 4: Verify the Result

Since \(1\) is greater than \(-\frac{1}{2}\), the inequality \(1 \geq -\frac{1}{2}\) is true. Therefore, the point \((1, 1)\) is included in the region described by the inequality.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points using a pair of numbers known as coordinates. Each point is identified by a unique pair \((x,y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. Think of it like a map where you can find locations based on these two values.

In this grid, there are four quadrants, created by two perpendicular lines called the axes:

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.

The point where these axes intersect is \((0, 0)\), known as the origin.

Understanding this plane is crucial for graphing inequalities and seeing where solutions are located. By plotting inequalities, we identify all points that make them true, thus determining the regions of solutions on this plane.

Graphing Inequalities

Graphing inequalities involves shading regions on the coordinate plane that satisfy the inequality. The first step is expressing the inequality in a standard form like \(y \geq -\frac{1}{2}x\).

Here's how you can graph inequalities:

• First, treat the inequality as an equation. So, graph the line \(y = -\frac{1}{2}x\). This line acts as a boundary for the inequality.
• If the inequality is \(\geq\) or \(\leq\), draw a solid line to include the boundary. For \(>\) or \(<\), use a dashed line.
• Determine which side of the line the inequality represents. You can do this by substituting a sample point (often \((0,0)\) if it isn't on the line), and seeing if it satisfies the inequality times \(x\)-coordinate value.
• Next, shade the region that includes all points satisfying the inequality. This shaded area represents all the potential solutions.

In our example, you’d see that the point \((1,1)\) falls on the shaded side of the boundary line.

Substitution Method

The substitution method is a straightforward way to verify if a particular point satisfies an inequality. It involves inserting the point's coordinates into the inequality to see if it holds true. Let's explore how it works:

Consider the inequality: \(y \geq -\frac{1}{2}x\).

• Given a point, such as \((1,1)\), substitute \(x = 1\) into the inequality.
• Similarly, substitute \(y = 1\) into the inequality.
• Compute and simplify: \(1 \geq -\frac{1}{2}\times 1\).
• Since \(1 \geq -\frac{1}{2}\) is true (because 1 is indeed greater), the point satisfies the inequality.

This method is invaluable for checking individual points when graphing on the coordinate plane. It confirms whether a point lies within the solution region indicated by the inequality on the graph.