The partners are given the literal inequality ax+b>c to solve for x. Joaquin says that he will solve it just like an equation. Serena says that he needs to be careful because if a is a negative number, the solution will be different. What do you say? What are the solutions for the inequality?
a x + b > c
Subtract b to both sides
a x + b - b > c - b
a x > c - b
If you multiply or divide both sides of an inequality by a negative number, reverse the direction of the inequality sign.
So if a > 0 then:
a x > c - b
If a < 0 then:
a x < c - b
Thank you, I appreciate the help!
a > 0
x > ( c - b ) / a
a < 0
x < ( c - b ) / a
ax > c - b
x > (c-b)/a
let's set some values for a, b, c, and x
of course our values have to satisfy the original
eg. x = 3, a = 5 , c = 10, b = 8
in original: 5(3) > 10-8 , true
in my statement: 3 > 2/5 , true
e.g. x = -5, a = 2, b = 20, c = 5
in original: 2(-5) + 20 > 5 , true
in mine: -5 > (5-20)/2 , -5 > -7.5 , true
x=-5 , a= -2, b= -20 c = -5
in original: -2(-5) - 20 > -5 , -10 > -5 , false
in mine: -5 > (-5+20)/-2 , -5 > 7.5 , still false
e.g. x = -5, a=2, b=20,c=-5
original: 2(-5) + 20 > -5 , true
mine: -5 > (-5-20)/2 , -5 > -7.5 , still true
what do you think ?
To solve the inequality ax + b > c for x, we need to remember a few key points. First, inequalities involving variables require special consideration since they represent a range of possible solutions rather than a single value. Second, we should keep in mind that if the coefficient 'a' is negative, it will affect the direction of the inequality.
To solve the inequality ax + b > c, we can start by subtracting 'b' from both sides of the inequality to isolate the term with 'ax'. This gives us ax > c - b.
Next, we need to consider the sign of 'a'. If 'a' is positive, the direction of the inequality remains the same when we divide both sides by 'a'. However, if 'a' is negative, dividing both sides by 'a' will flip the inequality.
So, to summarize:
1. If 'a' is positive:
- Divide both sides of ax > c - b by 'a' to get x > (c - b) / a.
2. If 'a' is negative:
- Divide both sides of ax > c - b by 'a' (which reverses the inequality sign) to get x < (c - b) / a.
Therefore, based on Joaquin's approach, he needs to remember to adjust the direction of the inequality if 'a' is negative before solving for x. Serena's caution is valid, as the solution will indeed be different when 'a' is negative compared to when it is positive.