Problem 42 - Algebra 1

Question

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

Step-by-Step Solution

Answer

Yes, the ordered pair (-2,20) is a solution to the inequality.

Step 1: Substitute for x and y

Substitute -2 for x and 20 for y in the inequality \(y \leq 2 x^{2}-3 x+10\). The inequality becomes: \(20 \leq 2 (-2)^{2}-3 (-2)+10\).

Step 2: Simplify the Right Side

Compute the right side: \(2 (-2)^{2}-3 (-2)+10 = 2 \cdot 4 + 6 + 10 = 24.\) The inequality now is \(20 \leq 24\).

Step 3: Verify the Inequality

As 20 is less than or equal to 24, hence, the inequality is true.

Key Concepts

Ordered Pairs

Ordered pairs are fundamental concepts in algebra and coordinate geometry. They consist of two elements where the order matters. Typically, an ordered pair is written in the format \((x, y)\), representing coordinates on the Cartesian plane.
The *first component* of an ordered pair corresponds to the x-coordinate, while the *second component* is the y-coordinate. These pairs help in identifying specific points in a two-dimensional space. In our exercise, the ordered pair \((-2, 20)\) signifies the point on the plane where the x-value is \(-2\) and the y-value is \(20\).
When solving problems involving inequalities or equations, verifying whether an ordered pair is a solution involves substituting the values into the given math expressions and checking the validity of the resulting statement. Each pair must satisfy the conditions specified by the inequality or equation to qualify as a solution.

Substitution

Substitution is a method used to simplify math expressions and verify solutions. In algebra, the principle of substitution involves replacing variables in an equation or inequality with given numbers to simplify computations or check results. This step plays a crucial role, allowing for numerical evaluation.
To apply substitution in our exercise, we replace \(x\) with \(-2\) and \(y\) with \(20\) in the inequality \(y \leq 2x^{2} - 3x + 10\). This converts the inequality to \(20 \leq 2(-2)^{2} - 3(-2) + 10\). Substitution enables us to transform and simplify complex algebraic expressions to evaluate the true or false nature of an inequality.
This method is straightforward and efficient, especially for confirming whether particular values make an inequality true. Ensuring the correct substitution and subsequent simplifications are crucial for obtaining valid results in mathematical tasks.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants connected by operations such as addition, subtraction, multiplication, and division. They form the backbone of algebra used to describe mathematical relationships and models.
In the inequality \(y \leq 2x^{2} - 3x + 10\), the expression \(2x^{2} - 3x + 10\) is an algebraic expression. It combines quadratic, linear, and constant terms:

• Quadratic term: \(2x^{2}\), where the variable \(x\) is squared.
• Linear term: \(-3x\), where the variable \(x\) is to the first power.
• Constant term: \(10\), with no variable attached.

These terms work together to represent the relationship dictated by the inequality.
Understanding how to manipulate algebraic expressions through operations like substitution or simplification is crucial for solving equations and inequalities. Mastery over these fundamentals allows students to approach more complex problems with confidence.