Problem 38 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the inequality. (Lesson 9.8) $$ y \leq 2 x^{2}-3 x+10,(-2,20) $$

Step-by-Step Solution

Answer

Yes, the ordered pair \((-2,20)\) is a solution to the inequality \(y \leq 2 x^{2}-3 x+10\).

Step 1 - Identifying the Ordered Pair

Firstly, we can identify that the ordered pair is \((-2,20)\). This implies that the \(x\) value is -2 and the \(y\) value is 20.

Step 2 - Substituting the Ordered Pair

Next, we substitute \(x = -2\) and \(y = 20\) in \(y \leq 2 x^{2}-3 x+10\).

Step 3 - Solving the Inequality

We obtain \(20 \leq 2 (-2)^{2} -3(-2) +10\), which simplifies to \(20 \leq 2*4 +6 +10 = 24\). Check if 20 is less than or equal to 24.

Step 4 - Analyzing the Result

The inequality \(20 \leq 24\) is true, therefore \((-2,20)\) is a solution to the inequality \(y \leq 2 x^{2}-3 x+10\).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair refers to a set of two elements organized in a specific sequence and enclosed in parentheses, such as \((-2, 20)\). The order of elements is crucial, as swapping the elements would result in a different ordered pair. We can think of an ordered pair as coordinates on a graph, where the first element represents the x-coordinate and the second element represents the y-coordinate.
This distinction helps locate a point in a two-dimensional plane. When considering ordered pairs in the context of inequalities, like in our exercise, they allow us to plug specific values into an inequality to assess whether they satisfy the condition posed by that inequality.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically formulated as \(ax^2 + bx + c\), where \(a, b,\) and \c\ are constants, with \(a eq 0\). Quadratic expressions are pivotal in formulations involving curves, as they represent parabola shapes on a graph.
A key property of quadratic expressions is their ability to open upward or downward. This is determined by the sign of the leading coefficient \(a\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
In the exercise, the given quadratic expression is part of an inequality: \(y \leq 2x^2 - 3x + 10\). This expression defines a range where the values of \y\ are less than or equal to a parabola described by the expression \(2x^2 - 3x + 10\). Analyzing quadratic expressions is crucial for understanding the solutions to related inequalities.
The calculations involve substituting a given ordered pair into the expression, which then allows us to see whether that pair lies within the constrained region of the inequality.

Solution of Inequalities

Inequalities involve comparing two expressions rather than testing for exact equality. A solution to an inequality determines whether a statement is true or false under certain conditions. In our example, we have the inequality \(y \leq 2x^2 - 3x + 10\). We check if the ordered pair \((-2, 20)\) satisfies this inequality.
To solve the inequality, substitute the ordered pair into the inequality's expression. Replace \(x\) with \(-2\) and \(y\) with \20\. This transforms the inequality into a true or false statement: \(20 \leq 2(-2)^2 - 3(-2) + 10\). After simplifying, if the resulting statement holds true, then the ordered pair is a solution.
In our case, the simplification leads to \(20 \leq 24\), which is true. Hence, the ordered pair is indeed a solution to the inequality.
Understanding how to break down and solve inequalities is important for applications in calculus, optimization problems, and real-world scenarios where boundaries are set by mathematical conditions.