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Flashcards: Inequalities & Absolute Value

$\displaystyle C=25h+5y$

The formula above gives the monthly cost, $\displaystyle C$, of running a food truck, where the driver works $\displaystyle h$ hours and $\displaystyle y$ gallons of gasoline are used for driving and operating the truck. If, in a particular month, it costs more than $3,500 to run the truck and no more than 220 gallons of gas were used, what is the minimum number of hours that the driver worked that month?

480

140

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Flashcards: Inequalities & Absolute Value

$\displaystyle C=25h+5y$

480

140

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Flashcards: Inequalities & Absolute Value

$\displaystyle C=25h+5y$

Your answer:

480

Your answer:

140

Your answer:

Correct answer:

Explanation:

You are given in this question that the equation C = 24h + 5y describes the cost to run a food truck and are asked to calculate the minimum number of hours the driver of the truck worked that month if the truck used no more than 220 gallons of gas. To find the minimum number of hours worked, you need to use the maximum number of gallons of gas that could have been used – 220. Substituting this value in for y gives you:

C = 25h + 5(220) = 25h + 1100

You are also told that it costs more than $3500 to run the truck for the month. You can then turn the equation into an inequality and solve for a range of values for h. Remember that saying that it costs at least $3500 is the same as saying that 3500 is less than or equal to the information given on the right side of the equation:

3500 < 25h + 1100

Subtracting 1100 from both sides and dividing both sides by 25 gives you:

2400 < 25h

96 < h

If h is is greater than or equal to 96, then the smallest number of hours worked must be 96.

Flashcards: Inequalities & Absolute Value

$\left | x+1 \right |>7$ $\displaystyle \left | x+1 \right |>7$

Which of the following provides the full solution set to the inequality above?

x>-6, x<8 $\displaystyle x>-6, x< 8$

x>6, x<7 $\displaystyle x>6, x< 7$

x>8, x<-6 $\displaystyle x>8, x< -6$

x>6, x<-8 $\displaystyle x>6, x< -8$

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Flashcards: Inequalities & Absolute Value

$\left | x+1 \right |>7$ $\displaystyle \left | x+1 \right |>7$

Which of the following provides the full solution set to the inequality above?

Your answer:

x>-6, x<8 $\displaystyle x>-6, x< 8$

Your answer:

x>6, x<7 $\displaystyle x>6, x< 7$

Your answer:

x>8, x<-6 $\displaystyle x>8, x< -6$

Correct answer:

x>6, x<-8 $\displaystyle x>6, x< -8$

Explanation:

When you're working with absolute values and inequalities, recognize that there are two cases that will satisfy the inequality: either the terms within the absolute value directly satisfy the inequality, or they satisfy the "opposite" (flip the sign and multiply by -1). For example, if $\left | a \right |>10$ $\displaystyle \left | a \right |>10$, then either a>10 $\displaystyle a>10$ or a<-10 $\displaystyle a< -10$.

For this problem, here is how the two cases will map out:

1) Satisfy the inequality directly:

x+1>7 $\displaystyle x+1>7$ means that x>6 $\displaystyle x>6$

2) Flip the inequality sign and multiply the numeric value by -1:

x+1<-7 $\displaystyle x+1< -7$, so x<-8 $\displaystyle x< -8$

This can feel abstract, so it's helpful to test one number for each case to see if it really works. For case 1, try a value greater than 6, so you might try 7. $\left | 7+1 \right |>7$ $\displaystyle \left | 7+1 \right |>7$ because 8 > 7, so you have this right. For case 2, try a number less than -8, such as -9. $\left | -9+1 \right |>7$ $\displaystyle \left | -9+1 \right |>7$ because the absolute value of -8 is 8, and 8 is greater than 7.

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