Interpretation of linear inequalities by graphical representation is the key element of understanding algebra. Graphs do not only make it easier to spot the solution set, they are also used to obtain significant data, like slope, y-intercept, and the kind of inequality due to the shading and the boundary lines. We should consider one of such examples below.
Question: Which Linear Inequality Is Represented by the Graph
Check the graph provided below. It presents a linear segment and also a gray area. According to the representation of the view, which linear inequality does this graph represent?
Options:
A. y>−2x+5y > -2x + 5y>−2x+5
B. y≤−2x+5y \leq -2x + 5y≤−2x+5
C. y≥−2x+5y \geq -2x + 5y≥−2x+5
D. y<−2x+5y < -2x + 5y<−2x+5
Correct Answer: B. y≤−2x+5y \leq -2x + 5y≤−2x+5
Explanation:
So then how can it be put simply? Step by step:
1. Line Type (Thin or Thick):
The graph depicts a strong thin line. In linear inequalities the solid line implies that the points on the line belong to the solution set. That would leave only B or C since they have the equal to subsets in (<= or >=).
2. Direction of Shading:
The dark part lies in the lower surface of the line. It implies a default less than kind of inequality, since the values of y within the area of the shade are not greater than the values on the line. Therefore the right answer should involve the use of symbol 1/2.
3. Line Equation:
The equation of the boundary line is written in slope-intercept form:
- y=mx+by=mx+by=mx+b
- where m represents the slope and b is y-intercept.
- The slope (m) here is -2 and the y-intercept (b) is 5, which means that we have the line
- y = -2x + 5 y = -2x + 5 = -2x +5
Given that the shaded area is below this line and the line is solid, the correct inequality is:
- y -2x+5y y <= -2x + 5y y -2x + 5
Thus, Option B is the correct answer.
Frequently Asked Questions (FAQs)
Q1: How can I tell if the inequality is “less than” or “greater than” from a graph?
A: The shaded region is a direction that indicates the nature of inequality. When the shading is below the line, the relationship is then less than (<) or smaller than or equal to (>) 1/2. If it is on the positive side, then there is greater than inequality (>) or greater than or equal to inequality (>=) inequality.
Q2: What does a solid line indicate in a linear inequality graph?
A: A solid line indicates that the solution set includes the line of boundary. This occurs when the inequality is = (less than or equal to) or = (greater than or equal to). In contrast, a dotted line indicates that the line is not part of it (when it comes to < or > inequalities).
