Problem 17 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the inequality. $$ y

Step-by-Step Solution

Answer

The ordered pair (-3,18) is not a solution to the inequality.

Step 1: Substitute the Values

The values from the ordered pair (-3,18) need to be substituted into the inequality. Replace 'x' with -3 and 'y' with 18. This results in \(18<-3^{2}+9(-3)\)

Step 2: Simplify the Inequality

Next, calculate the expression on the right side of inequality. This results in \(18< -3+(-27)\), which simplifies to \(18<-30\)

Step 3: Evaluate the Inequality

Finally, check if the inequality holds true or not. It's clear to see that 18 is not less than -30, hence the inequality doesn't hold.

Key Concepts

Ordered Pairs

An ordered pair is a pair of numbers used to locate a point on a coordinate plane, which is defined by its position along the horizontal (x-axis) and vertical (y-axis) axes.
For example, an ordered pair \( (x, y) \) consists of a first element \( x \) which represents the horizontal position, and a second element \( y \) which represents the vertical position.

An ordered pair can represent various mathematical entities, such as a point in geometry or a solution to an equation or inequality. In the context of inequalities, an ordered pair is considered a solution if, when the values are substituted into the inequality, the inequality holds true.

Let's consider the ordered pair \( (-3,18) \) and the inequality \( y < x^{2} + 9x \). When substituting \( x = -3 \) and \( y = 18 \) into the inequality, we’re effectively checking the ordered pair's location relative to the region described by the inequality on a coordinate plane. If the substitution results in a true statement, the ordered pair lies within that region; if not, it lies outside of it.

Solving Inequalities

Solving inequalities is about finding the values that make the inequality a true statement. Unlike equations, where we seek equality, inequalities show a relationship where one side is greater than, lesser than, or equal within a range when compared to the other side.

To solve an inequality, similar steps to those used in solving equations are taken: substituting values, simplifying expressions, and performing arithmetic operations. However, one critical difference is that when multiplying or dividing by a negative number, the inequality sign must be flipped.

Here's a straightforward process for solving inequalities:

• Substitute variables with given numbers if any.
• Simplify both sides of the inequality separately as needed.
• Perform any operations to isolate the variable, being mindful of the rules concerning multiplying or dividing by negative numbers.
• Check your solution by replacing the variable in the original inequality to verify that it makes a true statement.

Using these steps, you’ll quickly find whether your values satisfy the inequality or not.

Quadratic Inequalities

Quadratic inequalities are similar to quadratic equations, but instead of finding exact solutions, we're looking for ranges of values that satisfy the inequality. A quadratic inequality can be written in the form \( ax^{2} + bx + c > 0 \) or \( ax^{2} + bx + c < 0 \), for example.

To solve a quadratic inequality, you can follow these steps:

• Rewrite the inequality in standard form, with zero on one side.
• Find the roots of the equivalent quadratic equation \( ax^{2} + bx + c = 0 \) using methods like factoring, completing the square, or the quadratic formula.
• Use the roots to determine the intervals that satisfy the inequality. This involves testing intervals between the roots and selecting those which make the inequality true.
• Clearly denote the solution set, considering whether the roots are inclusive (using \( \geq \) or \( \leq \) when appropriate) or exclusive (using \( > \) or \( < \) when appropriate).

In the example \( y < x^{2} + 9x \), we're dealing with an inequality where the quadratic expression is set against a variable \( y \) instead of zero. However, the principles of solving remain conceptually consistent. Finding the range of x-values that satisfy the inequality relative to \( y \) gives us the solution set of the quadratic inequality.