Solve The Inequality: $\[ 4x - 5y \ \textgreater \ -5 \\]
Introduction
Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves finding the values of variables that satisfy the given inequality. In this article, we will focus on solving the inequality 4x - 5y > -5.
Understanding the Inequality
The given inequality is 4x - 5y > -5. To solve this inequality, we need to isolate the variables x and y. The inequality is already in a simple form, with the coefficients of x and y being integers.
Step 1: Isolate the Variable x
To isolate the variable x, we need to get rid of the term -5y. We can do this by adding 5y to both sides of the inequality.
4x - 5y > -5
4x > -5 + 5y
Step 2: Divide by the Coefficient of x
Next, we need to divide both sides of the inequality by the coefficient of x, which is 4.
4x > -5 + 5y
x > (-5 + 5y) / 4
Step 3: Simplify the Right-Hand Side
We can simplify the right-hand side of the inequality by combining the constants.
x > (-5 + 5y) / 4
x > -5/4 + 5y/4
Step 4: Write the Solution in Interval Notation
The solution to the inequality is x > -5/4 + 5y/4. We can write this in interval notation as (-∞, -5/4 + 5y/4).
Conclusion
Solving the inequality 4x - 5y > -5 involves isolating the variable x and then dividing by the coefficient of x. The solution to the inequality is x > -5/4 + 5y/4, which can be written in interval notation as (-∞, -5/4 + 5y/4).
Example Problems
Problem 1
Solve the inequality 3x - 2y > 2.
Solution
To solve this inequality, we need to isolate the variable x. We can do this by adding 2y to both sides of the inequality.
3x - 2y > 2
3x > 2 + 2y
Next, we need to divide both sides of the inequality by the coefficient of x, which is 3.
3x > 2 + 2y
x > (2 + 2y) / 3
We can simplify the right-hand side of the inequality by combining the constants.
x > (2 + 2y) / 3
x > 2/3 + 2y/3
The solution to the inequality is x > 2/3 + 2y/3, which can be written in interval notation as (-∞, 2/3 + 2y/3).
Problem 2
Solve the inequality 2x + 3y > 5.
Solution
To solve this inequality, we need to isolate the variable x. We can do this by subtracting 3y from both sides of the inequality.
2x + 3y > 5
2x > 5 - 3y
Next, we need to divide both sides of the inequality by the coefficient of x, which is 2.
2x > 5 - 3y
x > (5 - 3y) / 2
We can simplify the right-hand side of the inequality by combining the constants.
x > (5 - 3y) / 2
x > 5/2 - 3y/2
The solution to the inequality is x > 5/2 - 3y/2, which can be written in interval notation as (-∞, 5/2 - 3y/2).
Tips and Tricks
- When solving inequalities, it's essential to isolate the variable by adding or subtracting the same value to both sides of the inequality.
- When dividing both sides of the inequality by a coefficient, make sure to divide both the constant and the variable by the same value.
- When simplifying the right-hand side of the inequality, combine the constants to get a simpler expression.
Common Mistakes to Avoid
- When solving inequalities, avoid making mistakes by adding or subtracting the wrong value to both sides of the inequality.
- When dividing both sides of the inequality by a coefficient, avoid making mistakes by dividing the constant and the variable by different values.
- When simplifying the right-hand side of the inequality, avoid making mistakes by not combining the constants.
Conclusion
Introduction
In the previous article, we discussed how to solve inequalities by isolating the variable and then dividing by the coefficient of the variable. In this article, we will answer some frequently asked questions about solving inequalities.
Q: What is the difference between solving an equation and solving an inequality?
A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the values of the variable that make the inequality true.
Q: How do I know which direction to go when solving an inequality?
A: When solving an inequality, you need to determine the direction of the inequality. If the inequality is greater than (>), you need to go in the positive direction. If the inequality is less than (<), you need to go in the negative direction.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + by > c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c > 0, where a, b, and c are constants.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the expression to determine the solution.
Q: What is the difference between a system of linear inequalities and a system of quadratic inequalities?
A: A system of linear inequalities is a set of linear inequalities that must be satisfied simultaneously. A system of quadratic inequalities is a set of quadratic inequalities that must be satisfied simultaneously.
Q: How do I solve a system of linear inequalities?
A: To solve a system of linear inequalities, you need to graph the inequalities on a coordinate plane and then find the region where the inequalities overlap.
Q: What is the difference between a linear programming problem and a quadratic programming problem?
A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear constraints. A quadratic programming problem is a problem that involves maximizing or minimizing a quadratic function subject to a set of quadratic constraints.
Q: How do I solve a linear programming problem?
A: To solve a linear programming problem, you need to use a linear programming algorithm such as the simplex method or the interior-point method.
Q: What is the difference between a convex function and a concave function?
A: A convex function is a function that is concave up, meaning that its graph is a curve that opens upwards. A concave function is a function that is concave down, meaning that its graph is a curve that opens downwards.
Q: How do I determine whether a function is convex or concave?
A: To determine whether a function is convex or concave, you need to examine its second derivative. If the second derivative is positive, the function is convex. If the second derivative is negative, the function is concave.
Q: What is the difference between a local maximum and a global maximum?
A: A local maximum is a maximum value of a function that occurs at a specific point in the domain of the function. A global maximum is a maximum value of a function that occurs at a specific point in the domain of the function and is the largest value of the function in the entire domain.
Q: How do I find the local maximum of a function?
A: To find the local maximum of a function, you need to find the critical points of the function and then examine the sign of the second derivative at each critical point.
Conclusion
Solving inequalities involves isolating the variable and then dividing by the coefficient of the variable. By following the steps outlined in this article, you can solve inequalities with ease. Remember to avoid common mistakes and to simplify the right-hand side of the inequality by combining the constants.