Question

If \(J K=58\) and \(G H=67-3 b\), what values of \(b\) make \(J K \geq G H\) ?

Short answer

Values of \(b\) that make \(JK \geq GH\) are \(b \geq 3\).

Step-by-step solution

Set up the inequality

We know that \(JK = 58\) and \(GH = 67 - 3b\). We want to find values of \(b\) such that \(JK \geq GH\). Therefore, we set up the inequality: \[58 \geq 67 - 3b.\]

Isolate terms with b

Subtract 67 from both sides of the inequality to focus on terms with \(b\): \[58 - 67 \geq -3b.\] This simplifies to \[-9 \geq -3b.\]

Solve for b

To solve for \(b\), divide each side of the inequality by \(-3\). Remember to reverse the inequality sign when dividing by a negative number: \[\frac{-9}{-3} \leq b.\] This simplifies to \[3 \leq b.\]

Express the solution

The inequality \(3 \leq b\) means that \(b\) must be greater than or equal to 3. Therefore, all values of \(b\) that are 3 or greater will satisfy the original inequality.

Key concepts

Algebraic expressions

Algebraic expressions are a fundamental part of algebra, involving numbers and variables combined using mathematical operations such as addition, subtraction, multiplication, and division. In our exercise, we have expressions like \(JK = 58\) and \(GH = 67 - 3b\). Both of these expressions represent quantities.

• \(JK = 58\) is a simple expression where \(JK\) is a constant.
• \(GH = 67 - 3b\) is a more complex expression because it involves a variable \(b\).

Variables like \(b\) are placeholders for numbers, allowing us to write equations and inequalities that show relationships between different quantities. Understanding how to manipulate these expressions is key to solving problems like the one given.

Inequality solving

Inequality solving involves finding the values of variables that make an inequality true. In our problem, we need to solve the inequality \(58 \geq 67 - 3b\) to find the values of \(b\) that satisfy this condition. To do this, we follow a series of algebraic steps:

• Set up the inequality based on the problem: \(58 \geq 67 - 3b\).
• Simplify the inequality by isolating terms with the variable \(b\) on one side.
• Rearrange and solve the simplified inequality, remembering to flip the inequality sign if dividing by a negative number.

Here, we subtract 67 from both sides, leading to \(-9 \geq -3b\), and then divide each side by \(-3\), remembering to switch the inequality sign, resulting in \(3 \leq b\). Thus, \(b\) should be greater than or equal to 3.

Mathematical reasoning

Mathematical reasoning is the critical thinking that allows us to come to logical conclusions based on given information. In solving inequalities, it requires understanding how operations on an inequality affect the outcome.

• First, set up the problem by interpreting the given expressions.
• Use logical steps to simplify and manipulate the expressions accurately.
• Ensure to apply operations correctly, such as reversing inequality signs when dividing by negative numbers.

Our problem demonstrates reasoning through recognizing that when dividing both sides of an inequality by a negative number, the inequality sign must change direction. This procedural awareness ensures correct solutions and helps avoid common errors, leading to the correct interpretation that \(b \geq 3\).