Describe how you know if an inequality is an “and” or an “or” problem. (use -3 < x+5 < 8). Also, how do you know if an absolute value question is an “and” or an “or” problem?
-3 < x+5 < 8
subtract 5, (subtraction does not affect the direction of the inequality)
-8 < x < 3
This pattern implies the word "and"
- it can also be read:
x is between -8 and 3
- to have "between -8 and 3" mean: greater than -8 OR less than 3 would cover the entire number line, and any x would work, but clearly that would be a contradiction, so ....
As to absolute value:
|stuff| < k , where k is a positive number
stuff < k and -stuff < k
stuff < k and stuff > -k
or -k < stuff < k ,
|stuff| > k
stuff > k OR -stuff > k
stuff > k OR stuff < -k
stuff < -k OR stuff > k
I used to have my students remember that
"greater" sounds like "greatOR", and once you have established one choice, the other one is obvious.
To determine whether an inequality is an "and" or an "or" problem, we need to consider the relationship between the different parts of the inequality.
Let's take the example inequality -3 < x + 5 < 8. This is a compound inequality that consists of two inequalities connected with the "and" operator.
We can break it down into two separate inequalities: -3 < x + 5 and x + 5 < 8.
If we solve the first inequality, we subtract 5 from both sides: -3 - 5 < x + 5 - 5, which simplifies to -8 < x.
Similarly, if we solve the second inequality, we subtract 5 from both sides: x + 5 - 5 < 8 - 5, which simplifies to x < 3.
Now, let's analyze the absolute value questions. Absolute value is denoted by two vertical bars, ||. An absolute value question can involve an "and" or an "or" relationship.
For example, consider the equation |x - 2| > 5. To determine whether this is an "and" or an "or" problem, we focus on the inequality sign.
If the inequality sign is "greater than" or "less than," as in this case (>|<), then it indicates an "or" relationship.
To solve this absolute value inequality, we need to split it into two separate inequalities: x - 2 > 5 and -(x - 2) > 5.
Solving the first inequality, we add 2 to both sides: x - 2 + 2 > 5 + 2, which simplifies to x > 7.
Solving the second inequality, we distribute the negative sign: -x + 2 > 5. Then, we subtract 2 from both sides: -x + 2 - 2 > 5 - 2, which simplifies to -x > 3. To maintain the inequality's direction, we divide both sides by -1, flipping the inequality sign: x < -3.
Therefore, the solution to the absolute value inequality is x < -3 or x > 7, using the "or" relationship.