Write The Inequality Whose Graph Is Given

To write the inequality whose graph is given, make sure to first understand what an inequality is and how it is represented on a coordinate plane. An inequality is a mathematical statement that compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). When graphed, inequalities define a region on the coordinate plane rather than just a line.

To determine the inequality from a graph, follow these steps:

1. Identify the Boundary Line: The first step is to identify the equation of the boundary line. This is usually given in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. If the line is solid, the inequality includes the boundary (≤ or ≥). If the line is dashed, the boundary is not included (< or >).

2. Determine the Slope and Y-Intercept: Find the slope (m) by selecting two points on the line and using the formula m = (y2 - y1)/(x2 - x1). The y-intercept (b) is the point where the line crosses the y-axis.

3. Check the Shading: Look at which side of the line is shaded. If the region above the line is shaded, the inequality is y > mx + b or y ≥ mx + b. If the region below the line is shaded, the inequality is y < mx + b or y ≤ mx + b Not complicated — just consistent..

4. Write the Inequality: Combine the information about the boundary line and the shading to write the inequality. As an example, if the line is y = 2x + 3 and the region above the line is shaded with a solid line, the inequality is y ≥ 2x + 3.

Let's consider a few examples to illustrate this process:

Example 1: Suppose you are given a graph with a solid line passing through the points (0, 1) and (2, 5). The region above the line is shaded And that's really what it comes down to. Which is the point..

• Step 1: Identify the boundary line. The line passes through (0, 1) and (2, 5).
• Step 2: Determine the slope and y-intercept. The slope m = (5 - 1)/(2 - 0) = 4/2 = 2. The y-intercept b = 1. So, the equation of the line is y = 2x + 1.
• Step 3: Check the shading. The region above the line is shaded, and the line is solid.
• Step 4: Write the inequality. Since the region above the line is shaded and the line is solid, the inequality is y ≥ 2x + 1.

Example 2: Consider a graph with a dashed line passing through the points (-1, 3) and (1, -1). The region below the line is shaded.

• Step 1: Identify the boundary line. The line passes through (-1, 3) and (1, -1).
• Step 2: Determine the slope and y-intercept. The slope m = (-1 - 3)/(1 - (-1)) = -4/2 = -2. To find the y-intercept, use the point-slope form: y - y1 = m(x - x1). Using point (-1, 3), we get y - 3 = -2(x - (-1)), which simplifies to y - 3 = -2x - 2, so y = -2x + 1. Thus, the equation of the line is y = -2x + 1.
• Step 3: Check the shading. The region below the line is shaded, and the line is dashed.
• Step 4: Write the inequality. Since the region below the line is shaded and the line is dashed, the inequality is y < -2x + 1.

Example 3: Suppose the graph shows a horizontal line at y = 4, with the region above the line shaded and the line solid Less friction, more output..

• Step 1: Identify the boundary line. The line is horizontal at y = 4.
• Step 2: Determine the slope and y-intercept. The slope m = 0 (since it's a horizontal line), and the y-intercept b = 4. So, the equation of the line is y = 4.
• Step 3: Check the shading. The region above the line is shaded, and the line is solid.
• Step 4: Write the inequality. Since the region above the line is shaded and the line is solid, the inequality is y ≥ 4.

Example 4: Consider a graph with a vertical line at x = -3, with the region to the right of the line shaded and the line dashed That's the whole idea..

• Step 1: Identify the boundary line. The line is vertical at x = -3.
• Step 2: Determine the slope and y-intercept. Vertical lines have an undefined slope, and there is no y-intercept. The equation of the line is x = -3.
• Step 3: Check the shading. The region to the right of the line is shaded, and the line is dashed.
• Step 4: Write the inequality. Since the region to the right of the line is shaded and the line is dashed, the inequality is x > -3.

To keep it short, to write the inequality whose graph is given, you need to identify the boundary line, determine its equation, check the shading, and then write the inequality based on whether the line is solid or dashed and which side is shaded. This process allows you to translate a visual representation into a mathematical statement, which is a valuable skill in algebra and graphing That's the part that actually makes a difference. Which is the point..

Extending the Concept: Multiple Inequalities and Real‑World Applications

When a single boundary line can be described by an inequality, the same principles apply when more than one inequality must be satisfied simultaneously. In such cases the solution set is the intersection of the individual half‑planes, often visualized as a polygonal region on the coordinate plane.

5. Graphing a System of Two Linear Inequalities

Suppose we are asked to graph the system

[ \begin{cases} y \le 3x - 2\ y > -x + 4 \end{cases} ]

Step 1 – Plot each boundary line.

• For (y = 3x - 2) draw a solid line because the inequality is “(\le)”. - For (y = -x + 4) draw a dashed line because the inequality is “(>)”.

Step 2 – Choose a test point for each line.
A convenient choice is the origin ((0,0)) when it does not lie on the line. - Plugging ((0,0)) into (y \le 3x - 2) yields (0 \le -2), which is false; therefore the half‑plane that contains the origin is not part of the solution for the first inequality. Shade the opposite side It's one of those things that adds up..

• Plugging ((0,0)) into (y > -x + 4) gives (0 > 4), also false; so the region containing the origin is excluded for the second inequality, and we shade the side that does satisfy the condition.

Step 3 – Locate the overlapping shaded area. The region where the two shaded portions intersect is the set of all points that satisfy both inequalities. This area is typically a convex polygon bounded by the two lines Nothing fancy..

Step 4 – Verify with a point inside the intersection.
Pick a point that appears to lie in the overlapped region, such as ((2,2)). - (2 \le 3(2)-2 \Rightarrow 2 \le 4) (true) And it works..

• (2 > -2 + 4 \Rightarrow 2 > 2) (false, so ((2,2)) is not in the solution).
  Try ((3,5)):
• (5 \le 3(3)-2 \Rightarrow 5 \le 7) (true).
• (5 > -3 + 4 \Rightarrow 5 > 1) (true).
  Thus ((3,5)) belongs to the solution set, confirming the correct overlap.

6. Inequalities with Non‑Linear Boundaries

The process remains conceptually similar when the boundary is curved. As an example, consider the inequality

[x^{2}+y^{2}\le 9. ]

• Boundary: The circle (x^{2}+y^{2}=9) (radius 3, centered at the origin).
• Line type: Since the inequality is “(\le)”, the circle is drawn solid.
• Shading: The interior of the circle satisfies the condition, so the region inside the curve is shaded.

If a second inequality such as (y \ge x) were added, the solution would be the portion of the disk that lies above the line (y=x). Graphically, you would first draw the circle, then the line (solid or dashed according to the inequality), and finally shade the region that meets all conditions.

7. Translating Real‑World Constraints into Inequalities

Many word problems naturally lead to systems of inequalities. Consider a small business that produces two types of widgets, (A) and (B). Suppose the following constraints hold:

1. Each widget of type (A) requires 2 labor hours, and each of type (B) requires 3 labor hours. The total labor available per day is at most 120 hours.
2. The profit per unit of (A) is $40, and per unit of (B) is $60. The business wants to earn at least $2,400 per day.
3. No more than 40 units of (A) can be produced daily due to storage limits.

Let (x) be the number of type‑(A) widgets and (y) the number of type‑(B) widgets produced each day. The constraints translate to:

[ \begin{cases} 2x + 3y \le 120 &\text{(labor)}\[2pt] 40x + 60y \ge 2400 &\text{(profit)}\[2pt] x \le 40 &\text{(storage)}\[2pt] x \ge 0,; y \ge 0