Salary is $25,000

Let x be the total sales, so that Commission is 0.07 of x, or 0.07x

Income is at least $39,000 or ≥ $39,000

                     [latex]25000 + 0.07x≥ 39000[/latex]                          Salary + Commission ≥ Income

                                                                        [latex]0.07x≥ 14000[/latex]                           Subtract 25,000 from 39,000

                       [latex]\frac{0.07x}{0.07}≥\frac{14000}{0.07}[/latex]                            Divide both sides by 0.07

[latex]x≥200000[/latex]

If Maria’s total sales are $200,000, she will earn $39,000. 

If her total sales are more than $200,000, she will earn more than $39,000.

Answer

Maria’s total sales must be at least $200,000 in order to earn a yearly income of at least $39,000.

The cost per day is $18.95

Let x be the miles driven, so the mileage charge is 0.19[latex]x[/latex]

Leslie cannot go over $50, so ≤ $50. 

Daily rate + Mileage ≤ $50

[latex]18.95 + 0.19x≤50[/latex]

Subtract 18.95 on both sides

[latex]0.19x≤31.05[/latex]

Divide by 0.19 on both sides

[latex]\frac{0.19x}{0.19}≤\frac{31.05}{0.19}[/latex]

[latex]x≤163.4210526...[/latex]

Round to the nearest mile

[latex]x≤163[/latex]

If Leslie drives 163 miles, she will spend $50 on the car rental. If Leslie drives less than 163 miles, she will spend less than $50 on the car rental.

Answer

In order to spend at most $50 on the car rental, Leslie must drive no more than 163 miles.

Special Consideration

What if the solution to our inequality had been [latex]x≤163.8210526...[/latex]?  Normally, in this case we would round our answer to 164.  However, let’s look at what happens if we substitute 164 for the variable in our inequality.

[latex]18.95 + 0.19(164)≤50[/latex]

[latex]18.95 + 31.16≤50[/latex]

[latex]50.11≤50[/latex]

This is a false statement.

Substituting [latex]x=164[/latex] into the inequality results in a false statement.  Therefore, 164 is not a solution to the inequality.  Because Leslie has a maximum of $50 to spend, we cannot round our solution up or the cost of the car rental will exceed $50.

The cost for the first rental company is

[latex]18.95+0.19x[/latex]

while the cost for the second rental company is

[latex].75x[/latex].

To determine the number of miles Leslie can drive so that the second company is cheaper, we solve the following inequality:

[latex]0.75x< 18.95+0.19x[/latex]

[latex]\hspace{-.14in}\underline{-0.19x \hspace{.64in}-0.19x}[/latex]

[latex]0.56x<18.95[/latex]

[latex]\frac{0.56x}{0.56}<\frac{18.95}{0.56}[/latex]

[latex]x<\frac{18.95}{0.56}[/latex]

Initially, [latex]\frac{18.95}{0.56}\approx 33.8[/latex].  If we round down, we determine that if Leslie drives up to 33 miles, the second rental company will cost less.

Answer

Leslie can drive up to 33 miles.

Recall that to find an average, add the items and then divide by the number of items. Do not forget to include the fourth test score, which we can denote by [latex]x[/latex].

[latex]\frac{85+65+82+x}{4}≥80[/latex]

Multiply both sides by 4 to clear the fraction

[latex]4\cdot\frac{85+65+82+x}{4}≥80\cdot4[/latex]

[latex]85+65+82+x≥320[/latex]

[latex]232+x≥320[/latex]

Subtract 232 from both sides

[latex]x≥88[/latex]

If the student earns an 88 on the fourth test, he will have an average of 80. If the student earns a score higher than 88, his grade will be higher than 80.

Answer

The student must get a minimum score of 88 on the fourth test to earn an average of at least 80.