Solving and Understanding the Inequality 2x - 5 ≤ 11
Introduction to Inequalities
In mathematics, inequalities are used to compare two values, often with the relationship being less than, greater than, less than or equal to, or greater than or equal to. These relationships are crucial in solving various real-world and theoretical problems. One of the simplest forms of inequality is a linear inequality, which can be solved using algebraic techniques. Let's delve into solving the inequality 2x - 5 ≤ 11 step by step.
Understanding and Solving the Inequality 2x - 5 ≤ 11
To solve the inequality 2x - 5 ≤ 11, we need to isolate the variable x on one side of the inequality. Here are the steps to achieve this:
Move the constant term -5 to the right side of the inequality. This is done by adding 5 to both sides of the inequality: 2x - 5 ≤ 11 2x - 5 5 ≤ 11 5 2x ≤ 16 Next, we need to isolate x. Since 2 is the coefficient of x, we divide both sides of the inequality by 2: 2x / 2 ≤ 16 / 2 x ≤ 8The solution to the inequality 2x - 5 ≤ 11 is x ≤ 8. This means that any value of x that is less than or equal to 8 will satisfy the inequality.
Solving the Inequality Graphically
To represent the solution x ≤ 8 graphically, we can use a number line. We will draw a line and place a closed circle at 8, indicating that 8 is included in the solution. Then, we shade the region to the left of 8, indicating that all values up to and including 8 are part of the solution set.
Practical Application of Inequalities
Inequalities, particularly linear inequalities, have numerous real-world applications. For example, in economics, inequalities can be used to set price ranges for goods, ensuring a minimum or maximum price is not exceeded. In engineering, inequalities are used to ensure that component sizes fall within safe and functional limits. Understanding how to solve inequalities is a fundamental skill in mathematics and its applications.
Conclusion
In summary, the solution to the inequality 2x - 5 ≤ 11 is x ≤ 8. By moving the constant term to the other side and isolating the variable, we arrive at the solution. This method can be applied to a wide range of inequalities, enhancing our problem-solving skills in mathematics and various real-world scenarios.
Additional Resources
For further exploration of inequalities and linear equations, consider the following resources:
Khan Academy - Solving Linear Inequalities Math is Fun - Inequalities Math Planet - Linear Inequalities