Question

If $3x-7\ge -1,$ what is the minimum value of $x?$

Short answer

The minimum value of $x$ is 2.

Step-by-step solution

Understanding the Inequality

The inequality to solve is given as $3x-7\ge -1$. The goal is to isolate $x$ and find its minimum value.

Add 7 to both sides

To begin isolating $x$, add 7 to both sides of the inequality: $$3x-7+7\ge -1+7$$ Simplifying this, we get: $$3x\ge 6$$

Divide by 3

To solve for $x$, divide both sides of the inequality by 3: $$\frac{3x}{3}\ge \frac{6}{3}$$ This simplifies to: $$x\ge 2$$ The inequality now shows that $x$ must be greater than or equal to 2.

Determine the Minimum Value

The minimum value that $x$ can take, while still satisfying the inequality, is 2.

Key concepts

solving inequalities

Solving inequalities is a fundamental tool in algebra that allows us to understand the range of possible values for a given variable. In an inequality, you are often given a statement where two expressions are compared using inequality signs such as $\ge$, $>$, $\le$, or $<$. The main goal is to isolate the variable on one side of the inequality to find its acceptable range.

Let's take the given inequality, $3x-7\ge -1$. Our strategy is to manipulate this expression to find all possible values of $x$ that make the inequality true. Here’s a simple step-by-step approach:

• Add or subtract terms on both sides of the inequality to start isolating the variable.
• Divide or multiply both sides of the inequality by the same number to solve for the variable.

Remember that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

By following these steps rigorously, you can systematically solve any linear inequality.

algebraic expressions

Algebraic expressions are combinations of variables, coefficients, and mathematical operations (addition, subtraction, multiplication, division) that represent a value. In the exercise $3x-7\ge -1$, the algebraic expression is $3x-7$.

To understand and solve inequalities involving algebraic expressions, it's crucial to properly handle the components within the expression:

• Coefficients: Numbers multiplying the variable (3 in this case).
• Constants: Numbers without variables (-7 in this case).
• Variables: Symbols representing unknown values (x in this case).

Our main target is to work with these elements, using basic algebraic operations to isolate and solve for the variable. This process involves simplifying the expression and ensuring proper inequality rules are applied to reach a clear solution. Simplifying an algebraic expression accurately is vital for solving any inequality problems successfully.

minimum value

The minimum value of a variable in an inequality problem represents the smallest number that makes the inequality statement true. For the given exercise, we determined that $x\ge 2$ from the inequality $3x-7\ge -1$. To find this minimum value:

• Isolate the variable using algebraic operations. In this case, solve $3x\ge 6$ after adding 7 to both sides.
• Then divide both sides by the coefficient of the variable (3 in this case) to find $x\ge 2$.

Understanding the minimum value is essential for application in real-world problems where constraints or limitations are defined by inequalities. For instance, if you are budgeting or managing resources, recognizing the minimum or maximum bounds ensures efficiency and accuracy in your calculations.

Emphasizing the notion of mathematical boundaries (like minimum values) as a core concept solidifies a student's comprehension of inequalities and their significance in practical scenarios.