Problem 43 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y>4 x^{2}-48 x+61 ;(1,17)$$

Step-by-Step Solution

Answer

No, the given pair (1,17) is not a solution to the given inequality since the inequality after substitution gives a false statement.

Step 1: Substitute Values

Substitute the x- and y-values from the ordered pair into the inequality: \(17 > 4(1)^2 - 48(1) + 61\).

Step 2: Simplify Right-hand Side

Calculate the right side of the inequality resulting in: \(17 > 4 - 48 + 61\). This simplifies to \(17 > 17\).

Step 3: Check Validity

Evaluate if the inequality holds true. The inequality \(17 > 17\) is false.

Key Concepts

Understanding Ordered Pairs

Ordered pairs, such as the pair (1,17) in our exercise, are a fundamental concept in algebra and coordinate geometry. They represent the position of points on the Cartesian plane, where the first number denotes the x-coordinate, and the second number denotes the y-coordinate. In the context of inequalities, an ordered pair is considered a solution if, when substituted into the inequality, the inequality remains valid.

An important thing to remember is that when substituting the values from an ordered pair into an inequality, the pair must maintain its order: the first value is always the x-coordinate, and the second is always the y-coordinate. Confusion in this step may lead to incorrect conclusions about whether the pair is indeed a solution.

Solving Quadratic Inequalities

When dealing with quadratic inequalities like y > 4x^2 - 48x + 61, the procedure to solve them resembles that of quadratic equations. However, instead of looking for exact values, we're seeking ranges of x values that will make the inequality true. One method of solving such inequalities is by finding the roots of the corresponding quadratic equation and using them to test intervals on a number line.

If we imagine the quadratic equation as a parabola on the coordinate plane, the task is to determine where the y-value of a point (x, y) is greater than the y-value on the parabola. This correlates to finding the sections of the number line where the inequality holds true. Always be aware that the solution to a quadratic inequality is often a range of values rather than a specific ordered pair.

The Substitution Method

One straightforward technique for solving inequalities is the substitution method. In this approach, specific values are substituted into an inequality to test if it holds true. Following the steps outlined in our solution: we substitute x = 1 and y = 17 in the inequality 17 > 4(1)^2 - 48(1) + 61. After simplifying, we compare the resulting numerical values to determine the truth value of the inequality.

It's vital to perform substitution accurately and simplify correctly. If the inequality does not hold after substitution, as in the given exercise, where we find that 17 is not greater than 17, the ordered pair is not a solution. Substitution is a handy method, particularly when checking specific potential solutions or when dealing with linear inequalities.