Question

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

Short answer

Yes, the ordered pair (-2,-4) is a solution of the inequality \(y \geq x^{2}+4x\).

Step-by-step solution

Analyze and Begin Decision Process

Analyzing the inequality, identify that there are two variables, y and x. The task is then to determine whether replacing x and y in the inequality with -2 and -4, respectively (our ordered pair), satisfies the inequality.

Substitute Ordered Pair into Inequality

Substitute the values from the ordered pair into the inequality. This means replacing y with -4 and x with -2, getting \(-4 \geq (-2)^2 + 4*(-2)\).

Solve the Inequality

Simplify the equation. Calculate the result of the equation as \(-4 \geq 4-8\) , which simplifies to \(-4 \geq -4\).

Verification

Lastly, verify whether the equation is true. In this case \(-4 \geq -4\) is true. So, this pair is indeed valid and satisfy the inequality

Key concepts

Ordered Pairs

When you're exploring equations and inequalities, one concept you will frequently encounter is that of ordered pairs. An ordered pair, typically written as \( (x, y) \) is a set of numbers that represent the coordinates of a point on a two-dimensional plane. The first number represents the x-coordinate (horizontal position) and the second represents the y-coordinate (vertical position). Ordered pairs are used to plot points and represent solutions to equations involving two variables.

For example, consider the ordered pair \( (-2, -4) \). It signifies a point that is located 2 units to the left of the origin (since the x-coordinate is negative) and 4 units below the origin on a coordinate plane. To determine if this ordered pair satisfies a given inequality, such as \( y \geq x^2 + 4x \), substitution of the x and y values into the inequality checks if the relationship holds true. This process can help visualize solutions or even comprehend the shape of the graph that an equation represents, such as a parabola for quadratic equations.

Quadratic Inequalities

Quadratic inequalities are mathematical expressions that involve a quadratic term (where the variable is squared) and an inequality sign (such as <, >, \leq, or \geq). A quadratic inequality might look something like \( y \geq x^2 + 4x \). Unlike linear inequalities, which can be depicted as straight lines on a graph, quadratic inequalities represent regions bounded by a parabola.

When solving a quadratic inequality, there are few key steps to follow. The first step is to bring all terms to one side of the inequality, resulting in a standard form like \( x^2 + bx + c \leq 0 \text{or} \geq 0 \). Then, it's important to find the critical points by setting the quadratic equation to zero and solving for x, which gives you the x-intercepts of the parabola. The next step involves using a sign chart or plotting points to determine where the function satisfies the inequality. Finally, the solution can be expressed as an interval (or intervals), which corresponds to the sections along the x-axis that satisfy the inequality.

Substitution Method

The substitution method is a technique used to solve systems of equations or to determine the validity of an ordered pair in an inequality. This method involves replacing variables with given numbers or expressions that are considered equivalent based on the context of the problem at hand. It's a straightforward method that allows you to simplify complex problems by reducing the number of variables you need to deal with at one time.

Take our original problem as an example: to verify if the ordered pair \( (-2, -4) \) is a solution to the inequality \( y \geq x^2 + 4x \), you would use substitution. Replace the 'x' in the inequality with -2 and the 'y' with -4 and then simplify the equation. If the substitution results in a true statement, such as \( -4 \geq -4 \), then the ordered pair satisfies the inequality. This method not only assists in solving inequalities but is also enormously helpful when dealing with simultaneous equations where one equation can be solved for one variable and then substituted into another.