Problem 61 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y>2 x^{2}-7 x-15 ;(2,5)$$

Step-by-Step Solution

Answer

Yes, the ordered pair (2,5) is a solution to the inequality \(y > 2x^{2} - 7x - 15\) since the inequality is satisfied.

Step 1: Identify the ordered pair

In the pair (2,5), the first number is the x-value and the second number is the y-value. So, \(x = 2\) and \(y = 5\).

Step 2: Substitute the values into the inequality

Substitute \(x = 2\) and \(y = 5\) into the inequality \(y > 2x^{2} - 7x - 15\). This gives us an expression \(5 > 2(2)^{2} - 7(2) - 15\). That simplifies to \(5 > 2*4 - 14 - 15\).

Step 3: Simplify the inequality

Now we simplify the expression on the right side of the inequality. That gives us \(5 > 8 - 14 - 15\). Thus we have \(5 > -21\).

Step 4: Check the inequality

Now we check if the inequality \(5 > -21\) is true. And yes, 5 is greater than -21. Hence the inequality is true.

Key Concepts

Understanding Ordered Pairs

When working with coordinate systems, an understanding of ordered pairs is fundamental. An ordered pair ewline, generally written in the format ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline (x, y) ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline , consists of two elements where the first element ewline represents the horizontal position (x-value) and the second element ewline represents the vertical position (y-value) in a two-dimensional space. The exercise provides the ordered pair ewline (2, 5) ewline , indicating the x-value is 2 and the y-value is 5.ewline
ewline One may visualize this on a graph, where the 'x' axis is the horizontal line and the 'y' axis is the vertical. The point where the lines corresponding to the x and y values meet is the graphical representation of the ordered pair. In many algebra and geometry problems, determining whether an ordered pair satisfies an equation or inequality is essential.ewline
ewline When using ordered pairs to solve problems, it's crucial to substitute the values correctly into the given formulas or equations. Incorrect substitutions can lead to wrong conclusions, thus it's a skill that requires practice and attention to detail.

The Process of Inequality Substitution

Inequality substitution is similar to substituting values into an equation but with the goal of determining if the inequality holds true. The process involves replacing the variable in an inequality with a given number to assess the relationship between two expressions.ewline
ewline For instance, the inequality in the exercise is ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline y > 2x^2 - 7x - 15ewline ewline . To evaluate it, we replace 'x' and 'y' with 2 and 5 respectively. Once substituted, we simplify the inequality to check the truthfulness of the relationship. The substitution process is the bedrock of verifying solutions, as it clearly shows whether an ordered pair satisfies the inequality or not.ewline
ewline It's important to conduct each step carefully, especially simplification, to avoid arithmetic mistakes that could lead to incorrect answers. Practicing substitution helps in enhancing problem-solving skills and accuracy.

Solving Quadratic Inequalities

Quadratic inequalities are expressions where the highest degree of the variable is two, in an inequality setting. An inequality like ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline y > 2x^2 - 7x - 15ewline ewline is considered a quadratic inequality since it includes a squared variable term.ewline
ewline The process of solving these inequalities involves finding values of 'x' that make the inequality true. One approach to solve them is by graphing, where we first sketch the corresponding quadratic equation's parabola and then identify the regions where the inequality is satisfied. Another approach includes using algebraic techniques, such as factoring, completing the square, or employing the quadratic formula to find the critical points which divide the number line into intervals, then testing these intervals to see where the inequality holds.ewline
ewline Understanding quadratic inequalities is essential as these appear in various branches of mathematics and applications in physics, engineering, economics, and more. Mastery of solving these inequalities enables students to tackle more complex mathematical problems with confidence.