How do you know if a value is a solution for an inequality? How is this different from determining if a value is a solution to an equation?
o If you replace the equal sign of an equation with an inequality sign, is there ever a time when the same value will be a solution to both the equation and the inequality?
o Write an inequality and provide a value that may or may not be a solution to the inequality. Consider responding to a classmate by determining whether or not the solution provided is a solution to the inequality. If the value he or she provides is a solution, provide a value that is not a solution. If the value is not a solution, provide a value that is a solution.
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a.
To solve 3x>3, proceed to solve 3x=3 to get x=1.
Examine the inequality, if x=1+something, would the inequality be satisfied? If so, as in the case of the example, then you conclude that the solution is x>1.
The above method is true for linear relations. For non-linear relations, such as x²+4x+4>0, the method will not work.
b.
If the inequality is given as ≤ or ≥, then the solution with the equality sign will satisfy both the equation and the inequality.
c.
Make an example, and post your results if you would like a check.
To determine if a value is a solution for an inequality, you need to substitute the value into the inequality and check if it satisfies the inequality or not.
For example, let's say we have the inequality 2x + 3 > 7. To check if x = 2 is a solution to this inequality, we substitute the value of x into the inequality and simplify:
2(2) + 3 > 7
4 + 3 > 7
7 > 7
Since 7 is not greater than 7, the statement is false. Therefore, x = 2 is not a solution to the inequality 2x + 3 > 7.
Determining if a value is a solution to an equation is different because equations have an equal sign, and we are checking if the value makes the equation true.
For example, let's consider the equation 2x + 3 = 7. To check if x = 2 is a solution to this equation, we substitute the value of x into the equation and simplify:
2(2) + 3 = 7
4 + 3 = 7
7 = 7
Since 7 is equal to 7, the statement is true. Therefore, x = 2 is a solution to the equation 2x + 3 = 7.
Now, if you replace the equal sign of an equation with an inequality sign, there can be situations where the same value is a solution to both the equation and the inequality. However, it is not always guaranteed, as seen in the above examples.
Let's write an inequality and check if a value is a solution or not:
Example: 3x - 5 ≥ 10 (inequality)
Let's check if x = 5 is a solution:
3(5) - 5 ≥ 10
15 - 5 ≥ 10
10 ≥ 10
Since 10 is equal to 10, the statement is true. Therefore, x = 5 is a solution to the inequality 3x - 5 ≥ 10.
Now, let's provide a value that is not a solution to this inequality. Let's try x = 2:
3(2) - 5 ≥ 10
6 - 5 ≥ 10
1 ≥ 10
Since 1 is not greater than or equal to 10, the statement is false. Therefore, x = 2 is not a solution to the inequality 3x - 5 ≥ 10.
Lastly, let's provide a value that is a solution to this inequality. Let's try x = 7:
3(7) - 5 ≥ 10
21 - 5 ≥ 10
16 ≥ 10
Since 16 is greater than or equal to 10, the statement is true. Therefore, x = 7 is a solution to the inequality 3x - 5 ≥ 10.