How do I solve this:
The absolute value of ax+b is less than or equal to c for the terms of x in terms of a, b, and c where a>0 and c>0
-Then they ask:
The absolute value of ax+b is more than or equal to c for the terms of x in terms of a, b, and c where a>0 and c>0
It's the same thing except the second one is more than...
To solve the first equation, which states that the absolute value of ax+b is less than or equal to c, you can follow these steps:
1. Write down the given inequality: |ax + b| ≤ c.
2. Remove the absolute value bars by creating two separate inequalities: ax + b ≤ c and -(ax + b) ≤ c.
3. Solve the first inequality: ax + b ≤ c.
- To isolate x, subtract b from both sides: ax ≤ c - b.
- Divide both sides by a (since a is positive): x ≤ (c - b) / a.
4. Solve the second inequality: -(ax + b) ≤ c.
- Multiply both sides by -1 to change the direction of the inequality: ax + b ≥ -c.
- To isolate x, subtract b from both sides: ax ≥ -c - b.
- Divide both sides by a (since a is positive): x ≥ (-c - b) / a.
So, the solution to the first equation is (-∞, (c - b)/a] ∪ [(-c - b)/a, ∞).
Now let's move on to the second equation, which states that the absolute value of ax+b is more than or equal to c. The process is very similar:
1. Write down the given inequality: |ax + b| ≥ c.
2. Create two separate inequalities by removing the absolute value bars: ax + b ≥ c and -(ax + b) ≥ c.
3. Solve the first inequality: ax + b ≥ c.
- To isolate x, subtract b from both sides: ax ≥ c - b.
- Divide both sides by a (since a is positive): x ≥ (c - b) / a.
4. Solve the second inequality: -(ax + b) ≥ c.
- Multiply both sides by -1 to change the direction of the inequality: ax + b ≤ -c.
- To isolate x, subtract b from both sides: ax ≤ -c - b.
- Divide both sides by a (since a is positive): x ≤ (-c - b) / a.
So, the solution to the second equation is [(-c - b)/a, ∞) ∪ (-∞, (c - b)/a].