Question

After two games of bowling, Brenda has a total score of \(475 .\) To win the tournament, she needs a total score of 684 or higher. Let \(x\) represent the score she needs for her third game to win the tournament. Write an inequality for \(x .\) What is the lowest score she can get for her third game and win the tournament?

Short answer

The lowest score Brenda can get in her third game and still win the tournament is 209.

Step-by-step solution

Identifying Variables

First, we comprehend the problem and identify the necessary variables. Brenda scores 475 after two games. The total score required to win is 684. We are asked to find \(x\), which represents the score Brenda needs for her third game to win, leading us to the inequality \(475 + x \geq 684\).

Rearranging the Inequality

Second, we rearrange the inequality to solve for \(x\). Therefore we subtract 475 from both sides of the inequality. The inequality becomes \(x \geq 684 - 475\).

Computing the Value

Third, we perform the subtraction operation, such that \(x \geq 209\). This means that Brenda needs to score 209 or more in her third game to win the tournament.

Key concepts

Solving Inequalities

Understanding how to solve inequalities is an essential skill in algebra. Inequalities are similar to equations, but instead of indicating that two expressions are equal, they show that one expression is greater or less than another. To solve an inequality like the one Brenda faces in her bowling tournament, we start by setting up the inequality based on the information given.

In Brenda's case, she has a current score of 475 and needs a total score of at least 684 to win. To find the minimum score (\(x\)) she needs in the third game, we set up the inequality as: \[\begin{equation} 475 + x \text{'\textgreater'} '= 684. \end{equation}\] The goal, then, is to isolate the variable (\(x\)) on one side of the inequality to determine the range of possible values it can take. This is done through the application of arithmetic operations while maintaining the balance of the inequality. Calculating the minimum score required, we subtract 475 from both sides to find \[\begin{equation} x \text{'\textgreater'} '= 209. \end{equation}\] This result indicates that any score 209 or higher will ensure a win for Brenda.

Inequality Notation

Inequality notation conveys the relationship between two values. For algebraic inequalities, we use symbols such as \(>\) (greater than), \(<\) (less than), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to). When saying \(x \geq 209\), we mean that \(x\) is greater than or equal to 209. It includes 209 as a viable solution, alongside all higher numbers.

Understanding this notation is crucial since it dictates the range of possible solutions to an inequality. In normal circumstances, we'd graph the solution on a number line, where Brenda's minimum acceptable score of 209 would be marked with a closed dot (indicating inclusion) and extend an arrow to the right, showing all numbers higher than 209 are also solutions.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks for solving algebraic inequalities. When solving inequalities, these operations are used just like they are with equations, with one notable exception: if we multiply or divide by a negative number, the inequality symbol must be flipped.

In the context of Brenda's bowling score, we used subtraction to isolate the variable on one side of the inequality. If the inequality involved multiplication or division with negative numbers, we would need to reverse the inequality symbol. Understanding how to properly apply arithmetic operations ensures that solving inequalities is systematic and accurate, leading to a correct solution.