Step-by-Step Guide to Solve a System of Linear Inequalities on a Coordinate Plane.

solutions of a system of linear inequalities

are all of the ordered pairs that satisfy all the linear inequalities in the system.

A system of linear inequalities involves multiple linear inequalities that need to be solved simultaneously. The solution set of a system of linear inequalities is the set of all possible values of the variables that satisfy all the given inequalities. Below is a detailed step-by-step method to solve a system of linear inequalities:

Step 1: Graph each inequality individually on the coordinate plane using the following steps:

a) Isolate the variable on one side of the inequality;

b) Draw the line corresponding to the inequality using the slope and y-intercept if it’s a linear inequality in two variables, or plot points if it’s a nonlinear inequality in two variables;

c) Determine which side of the line (above or below) is included or excluded from the solution set.

Step 2: Identify the overlap regions by shading each region that satisfies one or more inequalities. The solution set is the intersection of all shaded regions. If there’s no overlap between the shaded regions, then the system has no solution.

Step 3: Write the solution set using set-builder or interval notation, depending on the type of variables involved:

a) If the variables are real numbers, use interval notation, e.g., (−∞, 3)∪(5, ∞);

b) If the variables are integers, use set-builder notation, e.g., {x|1 ≤ x ≤ 8, x ∈ Z}.

For example, consider the following system of linear inequalities:

3x + 4y ≥ 12
x − y < 5 y > x

Step 1: Graph each inequality on the coordinate plane:

a) 3x + 4y ≥ 12: y ≤ −3/4x + 3
b) x − y < 5: y > x − 5
c) y > x: y = x (draw a dotted line)

Step 2: Identify the overlap regions by shading:

The shaded region is where all three inequalities are satisfied.

Step 3: Write the solution set in interval notation:

S = {(x, y) | y ≤ −3/4x + 3, y > x − 5, y > x }

Therefore, the solution set is S =(−∞, ∞) × (5, ∞), or {(x, y) | x ∈ R, y > x, y > x − 5}.

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