Ways to make a number using subtraction

Learning Objectives Translate verbal phrases about subtraction into two-variable linear inequalities. Identify and graph the boundary line for a subtraction-based linear inequality. Correctly determine whether to use a solid or dashed line for the boundary. Use a test point to identify the correct half-plane representing the solution set. Represent all possible 'ways to make a number' that satisfy a subtraction condition by graphing the solution set on a coordinate plane. Determine if a given ordered pair (x, y) is a valid solution to a subtraction-based inequality. Model and interpret simple real-world scenarios involving a difference between two quantities. Ever wondered how many different scores you and a friend could have in a game where you're winning b...

TermDefinitionExample Linear Inequality in Two VariablesA mathematical statement that compares two expressions involving two variables (like x and y) using an inequality symbol (<, >, ≤, ≥). The graph of its solution is a region on the coordinate plane.x - y ≤ 10 is a linear inequality. It represents all pairs of numbers (x, y) where their difference is less than or equal to 10. Solution of an InequalityAn ordered pair (x, y) that makes the inequality a true statement when its values are substituted for the variables.For the inequality x - y > 4, the ordered pair (10, 3) is a solution because 10 - 3 = 7, and 7 is greater than 4. Boundary LineThe line that separates the coordinate plane into two half-planes. It is derived from the inequality by replacing the inequality symbol with...

Subtraction-Based Inequality Forms x - y < k, x - y > k, x - y ≤ k, x - y ≥ k These are the core forms we use to model situations where the difference between two quantities (x and y) is compared to a constant value (k). Boundary Line Equation To find the boundary line, replace the inequality symbol with an equals sign: x - y = k This equation gives you the line that divides the coordinate plane. This is the first step in graphing any two-variable inequality. Slope-Intercept Form Conversion x - y = k → -y = -x + k → y = x - k Rearranging the boundary line equation into slope-intercept form (y = mx + b) makes it much easier to graph. In this case, the slope 'm' is 1 and the y-intercept 'b' is -k.