Problem 60 - Algebra 1

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: Identify the ordered pair

The ordered pair given is \((-1,2)\). In this order, the first number denotes the 'x'-value and the second number denotes the 'y'-value.

Step 2: Substitute the values into the inequality

Substitute the value of x as -1 and y as 2 into the inequality. The inequality then becomes \(2 \leq (-1)^{2} - 7*(-1) + 9\).

Step 3: Simplify the inequality

Solving the right side of the inequality we get \(2 \leq 1 + 7 + 9\) which simplifies to \(2 \leq 17\).

Step 4: Verify the inequality

It is obvious that 2 is less than or equal to 17. Thus, the inequality \(2 \leq 17\) is true, which confirms that the given ordered pair \((-1,2)\) is a solution to the original inequality.

Key Concepts

Ordered Pairs

Understanding ordered pairs is fundamental when dealing with various mathematical concepts, including algebra and geometry. An ordered pair, commonly represented as \( (x, y) \), consists of two elements where the first element represents the horizontal position, and the second element indicates the vertical position on a coordinate plane.

Ordered pairs are essential when determining the location of points on graphs. In the context of inequalities, an ordered pair represents the 'x' and 'y' values that we plug into an equation or inequality to verify if the pair satisfies the given condition. For instance, if we have the ordered pair \( (-1, 2) \), -1 is the 'x' value, and 2 is the 'y' value. To determine if this pair is a solution to the inequality \( y \leq x^{2} - 7x + 9 \), we must substitute these values into the inequality, as demonstrated in the step-by-step solution.

Substitution Method

The substitution method is a staple mathematical technique used to solve systems of equations and also to check if an ordered pair satisfies an inequality. This method involves replacing the variables in an equation or inequality with the numbers from an ordered pair.

For example, when deciding if the ordered pair \( (-1, 2) \) is a solution to the inequality \( y \leq x^{2} - 7x + 9 \), we substitute \( x = -1 \) and \( y = 2 \) into the inequality, leading to \( 2 \leq (-1)^{2} - 7(-1) + 9 \). After the substitution, the inequality is then simplified to check if it holds true, which helps us verify the solution. Step-by-step substitution is a powerful tool to systematically solve problems and validate solutions.

Quadratic Inequalities

Quadratic inequalities are mathematical expressions that involve a quadratic polynomial and an inequality sign - such as \( <, >, \leq, \geq \). They can often be solved graphically by plotting the corresponding quadratic equation and identifying the regions where the inequality is satisfied.

Take the inequality \( y \leq x^{2} - 7x + 9 \). To solve it graphically, we could plot the equation \( y = x^{2} - 7x + 9 \) and look for the areas on the graph where the 'y' values are less than or equal to the 'y' values on the curve. Algebraically, we can determine specific ordered pairs that satisfy the inequality, meaning that when we substitute the 'x' and 'y' values from the pair into the inequality, it makes a true statement, as seen when we confirmed that \( (-1, 2) \) was a solution. Understanding how to solve quadratic inequalities helps in various fields such as engineering, physics, and economics where such conditions need to be optimized.