5. working with inequalities is similar to working with equations. a solution to a one-variable inequality is a value for the variable that makes the inequality true. the set of all solutions to an inequality is a set of numbers that all make the inequality true. the solution to a one-variable inequality is usually graphed on a number line. most of the properties of equality can also be used to solve inequalities. the only exception is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign to produce an equivalent inequality. solve the inequality. (a) solve the inequality: . (b) the manager at super fit is not adding members at the rate he wants so he decides to change his fee structure to see if that . his new fee structure reduces the one-time registration fee to $24 and raises the monthly fee to $24 a month. solve the inequality to determine how many months it will take for a member to be paying more under the new plan than the old plan.

A

Should there be some inequality to solve for A? I'll assume not unless you post an edit.

B

We have a problem here as well. We don't know what the old fee structure was. Suppose I just use symbols.

Old cost for one time membership = y

Old monthly fee = x

Number of months = m

x < 24; y > 24

old income < new income

y +  xm < 24 + m*24

(y  - 24) < m*(24 - x)

(y - 24)/(24 - x) < m   The monthly fee went up the registration went down. There was no change in sign for division.  M is determined by how big the denominator is on the left. If the change is not much m will take a while to be greater than 1.

Edit

First Equation

5x + 2(x + 2) ≤ - 10       Remove the brackets

5x + 2x + 4 ≤ - 10         Collect the like terms on the left

7x + 4 ≤ - 10                  Subtract 4 from both sides

7x ≤ - 10 - 4

7x ≤ - 14                         Divide by 7

x ≤ - 14/7

x ≤ - 2            The number line would start with a filled in dot on - 2 and go left from there.

Second inequality

24 + 24x ≥ 150 + 10x        Subtract 24 from both sides.

24x ≥ 150 - 24 + 10x         Collect like terms on the right  

24x ≥ 126 + 10 x                Subtract 10x  from both sides

24x - 10x ≥ 126                  Collect like terms on the left  

14x ≥ 126                            Divide by 14

x ≥ 126 / 14

x ≥ 9

5

step-by-step explanation: