Problem 16 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y \geq x^{2}-13 x,(-1,14)$$

Step-by-Step Solution

Answer

Yes, the ordered pair (-1,14) is a solution to the inequality.

Step 1: Substitution of values

Replace x with -1 and y with 14 in the inequality, which becomes: 14 \geq (-1)^{2}-13*(-1)

Step 2: Simplify the inequality

Solve the inequality. The left-hand side (LHS) of the inequality retains its value, that is 14. The right hand side (RHS) simplifies to 1 + 13, which equals 14.

Step 3: Comparison

Compare the LHS and RHS of the inequality. This results in: 14 \geq 14.

Key Concepts

Ordered Pair

An ordered pair is a fundamental concept in mathematics, particularly in coordinate geometry. It is a set of numbers used to locate the position of a point on a graph, and it is always expressed in the form \( (x, y) \), where \( x \) is the horizontal component known as the abscissa, and \( y \) is the vertical component called the ordinate.

In the context of solving inequalities, an ordered pair represents a potential solution, where the \( x \) and \( y \) values are substituted into the inequality to verify if the relationship holds true. It's important to check each part of the ordered pair meticulously to confirm if the pair satisfies the inequality.

Quadratic Inequality

Quadratic inequalities are mathematical expressions that compare a quadratic function to a constant value using inequality symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). A quadratic function is of the form \( y = ax^2 + bx + c \), where \( a \) is not equal to zero.

The solution to a quadratic inequality involves finding the range of \( x \) values that make the inequality true. To verify a potential solution, one must substitute the \( x \) value into the quadratic equation and compare the calculated \( y \) value against the \( y \) component of the ordered pair or the constant that the quadratic is set against. In the case of \( y \geq x^2 - 13x \) with the ordered pair \( (-1, 14) \), checking the inequality involves ensuring that the right side of the inequality, when evaluated at \( -1 \) for \( x \) does not exceed the \( y \) value.

Substitution Method

The substitution method is a technique used in algebra to solve equations and inequalities. It involves replacing variables with their corresponding values to simplify the equation or inequality and solve for the unknown. Regarding inequalities, this means substituting the given values of \( x \) and \( y \) into the inequality and then checking to see if the statement is true.

To check an ordered pair as the solution to a quadratic inequality, substitute the \( x \) and \( y \) values into the inequality and simplify. If both sides of the inequality remain true after substitution, then the ordered pair is indeed a solution. For instance, in the given problem, substituting \( x = -1 \) and \( y = 14 \) into the inequality, we simplify and compare both sides to verify that the inequality holds. This method offers a systematic approach to validating solutions for inequalities.