Problem 47 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y \geq-x^{2}+3 x-\frac{15}{4} ;(2,-3)$$

Step-by-Step Solution

Answer

No, the ordered pair (2,-3) is not a solution to the given inequality \(y \geq -x^{2}+3x-\frac{15}{4}\).

Step 1: Identify the inequality equation and the ordered pair

The inequality that we are working with is \(y \geq -x^{2}+3x-\frac{15}{4}\) and the ordered pair to check is (2,-3). The first number 2 is the x-coordinate, and -3 is the y-coordinate.

Step 2: Substitute the coordinates of the ordered pair into the inequality equation

Substitute 2 for x, and -3 for y, which gives: \(-3 \geq - (2)^2 + 3*(2) - \frac{15}{4}\). This simplifies to, \(-3 \geq -4 + 6 - \frac{15}{4}\).

Step 3: Simplify the inequality equation and check the validity

Further simplifying the inequality we get \(-3 \geq 2 - \frac{15}{4} = - \frac{7}{4}\). Since -3 is not greater than or equal to - \frac{7}{4}, the ordered pair (2,-3) is not a solution to the given inequality.

Key Concepts

Ordered Pairs

Understanding ordered pairs is crucial when dealing with graphical representations of equations and inequalities. An ordered pair, written as \( (x, y) \), consists of two elements where the first element represents the horizontal coordinate, or 'x-coordinate', and the second element represents the vertical coordinate, or 'y-coordinate'. These coordinates correspond to a point on the Cartesian plane. To determine if an ordered pair is a solution to an inequality, you substitute the x and y values into the inequality and check if the inequality holds true.

For example, given the ordered pair (2, -3) and the inequality \( y \geq -x^2 + 3x - \frac{15}{4} \), we would replace 'x' with 2 and 'y' with -3. If the resulting expression is true, then the ordered pair is a solution to the inequality. The interpretation of ordered pairs is a foundational skill for graphing and solving inequalities, as it provides a clear method to visually verify solutions on a graph.

Quadratic Inequalities

Quadratic inequalities are mathematical expressions that involve a quadratic expression on one side of the inequality symbol. They are of the form \( ax^2 + bx + c < 0 \) or \( ax^2 + bx + c \leq 0 \), where '<' stands for 'less than' and '\leq' stands for 'less than or equal to'. The process of solving a quadratic inequality involves finding the values of 'x' that make the inequality true.

To solve a quadratic inequality, you can first solve the corresponding quadratic equation \( ax^2 + bx + c = 0 \) to find its roots. The roots split the number line into intervals. By selecting a test point from each interval and substituting it into the original inequality, you can determine which intervals satisfy the inequality. These intervals are the solution set to the inequality. Visualizing the inequality on a graph can also aid in understanding the regions where the inequality is true, with the parabola formed by the quadratic equation acting as the boundary.

Substitution Method

The substitution method is a technique used to solve systems of equations where you solve one equation for one variable and then substitute the result into another equation. This method can also be used to check solutions to inequalities.

To apply the substitution method, follow these steps:

Identify the Variable to Substitute

Choose a variable from one equation that you can easily express in terms of the other variable.

Solve for the Variable

Isolate the chosen variable on one side of the equation.

Substitute into the Other Equation

Replace the isolated variable in the second equation with the expression obtained from the first equation. Solve the resulting equation. For inequalities, you will substitute the values directly without solving for the variable first, and then determine if the inequality holds true for those values. This method enables you to plug in and check the validity of a potential solution quickly and efficiently.