Problem 54 - Algebra 1: Concepts and Skills

Question

Decide whether the ordered pair is a solution of the inequality. $$ y>x^{2}-2 x+5 ;(1,-7) $$

Step-by-Step Solution

Answer

No, the ordered pair (1, -7) is not a solution of the inequality \(y > x^{2} - 2x + 5\).

Step 1: Substitute values

First, substitute the x value (1) and y value (-7) from the ordered pair into the inequality. The inequality will now be \(-7 > 1^{2} - 2*1 + 5\)

Step 2: Simplify the inequality

Next, simplify the right-hand side of the inequality. This gives \(-7 > 1 - 2 + 5\), which simplifies further to \(-7 > 4\)

Step 3: Evaluate the inequality

Finally, observe the inequality. The statement \(-7 > 4\) is not true. Therefore, the ordered pair (1, -7) is not a solution of the given inequality.

Key Concepts

Ordered Pairs

An ordered pair is a pair of elements written in a specific sequence, generally expressed as \(x, y\). These elements often represent coordinates on a plane, where 'x' denotes the horizontal position and 'y' represents the vertical position.
It's essential in graphing and solving equations, especially inequalities, as it pinpoints exact positions within a graph or a mathematical equation.
In the context of inequalities, determining if an ordered pair, such as (1, -7), is a solution involves checking if these values satisfy the inequality. The order of elements must not be altered; 'x' always comes before 'y'.
If the pair, when plugged into the inequality, results in a true statement, it is considered a solution. If not, as in our problem where \(-7 > 4\) is false, the ordered pair is not a solution.

Substitution in Inequalities

When dealing with inequalities, substitution is the process of replacing the variable(s) in the inequality with specific values from an ordered pair. This approach helps to determine whether the given pair is a valid solution to the inequality.
Here's how you do it:

• Identify the values in the ordered pair, labeled as \(x\) and \(y\).
• Replace the variable \(x\) in the inequality with the given \(x\) value.
• Similarly, substitute the variable \(y\) with the given \(y\) value.

For instance, in the inequality \(y > x^2 - 2x + 5\), substitution with the pair \( (1, -7) \) transforms it into \(-7 > 1^2 - 2 \cdot 1 + 5\). The next step involves simplifying this transformed inequality to check its truth value.

Evaluating Inequality Expressions

After substitution, the next step in solving an inequality is evaluating the expression.
Evaluation involves simplifying and making sense of the inequality to see if the statement holds true or false. To do this:

• First, perform any calculations on the variables and constants.
• Check whether the final left-hand value is greater, less, or equal to the right-hand value according to the inequality sign.

Using our example, simplify \(-7 > 1 - 2 + 5\) by calculating each term on the right. The expression \(1^2 - 2 + 5\) simplifies to \(4\), leading us to the inequality \(-7 > 4\).
Since \(-7\) is not greater than \(4\), the inequality doesn’t hold true, confirming that the ordered pair \( (1, -7) \) is not a solution.