System of Inequalities Point Checker
Enter the coordinates of a point (x, y) and up to two linear inequalities to check if the point satisfies the system.
<input type="text" id="inequality1" value="2x + 3y
= 1″>
Evaluation for Point (" + x + ", " + y + "):
"; var allSatisfied = true; var atLeastOneInequality = false; // Evaluate Inequality 1 if (inequality1String.trim() !== "") { atLeastOneInequality = true; var eval1 = evaluateInequality(ineality1String, x, y); if (!eval1.success) { resultDiv.innerHTML = "Error in Inequality 1: " + eval1.message + ""; return; } outputHtml += "Inequality 1 (" + inequality1String + "): " +
(eval1.result ? "Satisfied" : "Not Satisfied") + "";
if (!eval1.result) {
allSatisfied = false;
}
} else {
outputHtml += "Inequality 1 was left blank.";
}
// Evaluate Inequality 2
if (inequality2String.trim() !== "") {
atLeastOneInequality = true;
var eval2 = evaluateInequality(inequality2String, x, y);
if (!eval2.success) {
resultDiv.innerHTML = "Error in Inequality 2: " + eval2.message + "";
return;
}
outputHtml += "Inequality 2 (" + inequality2String + "): " +
(eval2.result ? "Satisfied" : "Not Satisfied") + "";
if (!eval2.result) {
allSatisfied = false;
}
} else {
outputHtml += "Inequality 2 was left blank.";
}
if (!atLeastOneInequality) {
outputHtml += "Please enter at least one inequality to check.";
allSatisfied = false; // No inequalities means no system to satisfy
} else if (allSatisfied) {
outputHtml += "Conclusion: The point (" + x + ", " + y + ") SATISFIES ALL inequalities in the system.";
} else {
outputHtml += "Conclusion: The point (" + x + ", " + y + ") DOES NOT SATISFY ALL inequalities in the system.";
}
resultDiv.innerHTML = outputHtml;
}
Understanding Systems of Inequalities
A system of inequalities consists of two or more inequalities with the same variables. The solution to a system of inequalities is the set of all points that satisfy every inequality in the system simultaneously. Graphically, this solution set is represented by the region where the shaded areas of all individual inequalities overlap.
How to Solve Systems of Inequalities Graphically
- Graph Each Inequality: For each inequality, first graph the corresponding linear equation (replace the inequality sign with an equals sign). This line acts as a boundary.
- Determine Line Type: If the inequality is strict (
<or>), the boundary line is dashed, indicating that points on the line are not part of the solution. If the inequality includes equality (<=or>=), the boundary line is solid, meaning points on the line are part of the solution. - Shade the Solution Region: Choose a test point (often (0,0) if it's not on the boundary line) and substitute its coordinates into the original inequality.
- If the test point satisfies the inequality, shade the region containing the test point.
- If the test point does not satisfy the inequality, shade the region opposite to the test point.
- Identify the Overlap: The solution to the system is the region where all the shaded areas from each inequality overlap. This overlapping region represents all the points (x, y) that satisfy every inequality in the system.
The System of Inequalities Point Checker
While graphing provides a visual solution, this calculator offers a quick way to verify if a specific point (x, y) lies within the solution set of a given system of inequalities. Instead of drawing graphs, you can input the coordinates of a point and up to two inequalities. The calculator will then evaluate each inequality at that point and tell you whether the point satisfies each one individually, and thus, the entire system.
How to Use This Calculator
- Enter X-coordinate: Input the numerical value for the 'x' variable of the point you want to check.
- Enter Y-coordinate: Input the numerical value for the 'y' variable of the point you want to check.
- Enter Inequality 1: Type your first inequality. Use standard mathematical notation.
- Supported operators:
<(less than),>(greater than),<=(less than or equal to),>=(greater than or equal to). - Examples:
2x + 3y < 10,x - y >= 1,x > 0,y <= 5. - Coefficients like
2xare automatically interpreted as2*x. For other multiplications (e.g.,x*yorx*2), please use the asterisk (*) explicitly.
- Supported operators:
- Enter Inequality 2 (Optional): Type your second inequality following the same rules. You can leave this blank if you only have one inequality.
- Click "Check Point": The calculator will process your inputs and display whether the point satisfies each inequality and the system as a whole.
Examples of Use
Example 1: Point Does Not Satisfy the System
Let's check if the point (x=1, y=2) satisfies the system:
2x + 3y < 10x - y >= 1
Input:
- X-coordinate:
1 - Y-coordinate:
2 - Inequality 1:
2x + 3y < 10 - Inequality 2:
x - y >= 1
Calculation:
- For
2x + 3y < 10: Substitute x=1, y=2:2(1) + 3(2) = 2 + 6 = 8. Since8 < 10is TRUE, the first inequality is satisfied. - For
x - y >= 1: Substitute x=1, y=2:1 - 2 = -1. Since-1 >= 1is FALSE, the second inequality is NOT satisfied.
Result: The point (1, 2) does not satisfy all inequalities in the system because the second inequality is not met.
Example 2: Point Does Not Satisfy the System (Another Case)
Let's check if the point (x=5, y=0) satisfies the system:
x + y <= 4x > 0
Input:
- X-coordinate:
5 - Y-coordinate:
0 - Inequality 1:
x + y <= 4 - Inequality 2:
x > 0
Calculation:
- For
x + y <= 4: Substitute x=5, y=0:5 + 0 = 5. Since5 <= 4is FALSE, the first inequality is NOT satisfied. - For
x > 0: Substitute x=5:5 > 0is TRUE, the second inequality is satisfied.
Result: The point (5, 0) does not satisfy all inequalities in the system because the first inequality is not met.
Example 3: Point Satisfies a Single Inequality
Let's check if the point (x=3, y=1) satisfies the inequality:
y < 2x - 1
Input:
- X-coordinate:
3 - Y-coordinate:
1 - Inequality 1:
y < 2x - 1 - Inequality 2: (Leave blank)
Calculation:
- For
y < 2x - 1: Substitute x=3, y=1:1 < 2(3) - 1which is1 < 6 - 1or1 < 5. This is TRUE.
Result: The point (3, 1) satisfies the inequality.