We are asked to explain the similarities and differences between a linear inequality in two variables and a linear equation in two variables. Let's first recall what each of them are and then make our conclusions.
Linear equation in two variables
A linear equation in two variables is a mathematical expression involving an equality where two variables are involved. Let's look at an example.
y = x +3
The graphic representation of this equation is a collection of points which satisfy the equality. In this case, it is a straight line.
Linear inequality in two variables
A linear inequality in two variables is obtained by replacing the equal sign in a linear equation with an inequality symbol, such as <, >, ≥, or ≤ .
Linear equation& Linear inequality
y = x +3 & ⇒ y < x +3
The graphic representation is a collection of points which satisfy the inequality. To graph it we first graph its boundary line, which can be obtained by replacing the inequality sign with an equality sign.
Linear inequality Boundary line
y < x +3 ⇒ y = x +3
If the inequality sign is < or > we will use a dotted line when graphing it. Meanwhile, if it is ≥ or ≤ we will use a solid line. In this case, we need to use a dotted line.
We can determine which region contains the solution set by using a test point. In this case, we will try (0,0).
We got a true statement, so we would shade the region containing the test point. Therefore, the solution to our inequality would be graphed as shown below.
Conclusion
Let's summarize the similarities and differences between a linear inequality in two variables and a linear equation in two variables.
Similarities: For what we saw above, we can see that both mathematical expressions are closely related. In fact, we can obtain a linear inequality in two variables just by replacing the equals sign in a linear equation in two variables with an inequality symbol.
Linear equation& Linear inequality
y = x +3 & ⇒ y < x +3
Differences: The solution set for a linear equation in two variables is represented graphically as a straight line, while we can only graph the boundary line for a linear inequality in two variables using only a line. To completely graph the linear inequality, we need to shade the region containing the points that satisfy it.
