Problem 52 - Algebra 1: Concepts and Skills

Question

Decide whether the ordered pair is a solution of the inequality. $$ y \leq x^{2}-7 x+9 ;(-1,2) $$

Step-by-Step Solution

Answer

Yes, the ordered pair (-1,2) is a solution to the inequality \(y \leq x^{2}-7x+9\).

Step 1: Write down the inequality

The inequality is \(y \leq x^{2}-7x+9\).

Step 2: Substitution

Substitute \(x = -1\) and \(y = 2\) into the inequality: \(2 \leq (-1)^{2}-7(-1)+9\). Calculate the right side of the inequality.

Step 3: Verification

After calculation, the right side of the inequality equals 17. Therefore, the inequality becomes \(2 \leq 17\), and since this is true, the ordered pair (-1,2) is indeed a solution to the inequality.

Key Concepts

Ordered Pair Solution

When working with inequalities, which are mathematical expressions using the symbols '<', '>', '\(\leq\)' (less than or equal to), and '\(\geq\)' (greater than or equal to), an ordered pair solution refers to a set of two numbers that can replace the variables in an inequality, resulting in a true statement.

Consider the simple inequality \(y \leq x^{2} - 7x + 9\). To determine if an ordered pair, such as \((-1, 2)\), is a solution, a method called substitution is used. In this context, '-1' is substituted for 'x' and '2' for 'y'. If the resulting inequality holds true after the substitution, then the pair is indeed a solution to the inequality. Ordered pairs are commonly plotted on a coordinate plane, and finding solutions to inequalities can be a crucial step when graphing the corresponding regions.

Quadratic Inequalities

A quadratic inequality is similar to a quadratic equation, but instead of an equals sign, it includes an inequality symbol. In other words, it involves an expression with an \(x^{2}\) term and takes the form \(ax^{2} + bx + c \) followed by '\(<\)', '\(>\)', '\(\leq\)', or '\(\geq\)'.

To solve a quadratic inequality, you typically need to find the roots of the corresponding quadratic equation and then use those roots to determine the intervals for which the inequality is true. Graphically, this often involves sketching the parabola to see which parts of the graph lie above or below the x-axis, corresponding to positive or negative values of 'y' (or the function). Understanding the behavior of the parabola is key to solving these types of inequalities.

Substitution Method

The substitution method is a technique used to solve systems of equations or inequalities. The idea is to replace a variable with a given number or another expression, making the calculation simpler and easier to manage. This method is especially useful when the equation or inequality is already solved for one variable in terms of another, or if the solution to a variable is already known, as in the case of testing an ordered pair.

For example, when deciding if \((-1,2)\) is a solution to the inequality \(y \leq x^{2} - 7x + 9\), the substitution method involves plugging '-1' into every instance of 'x' and '2' into 'y'. This direct substitution simplifies the inequality and allows for immediate verification. It helps to visualize the task as replacing a placeholder with a concrete number, to see if the final, reduced inequality is true or false.