Question

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a. $2x + 3leq x + 5$
b. $-2x + 3>x + 9$
c. $27geq 5x + 4x$
d. $-8x + 2x - 16< -5x + 7x$
e. $3(x - 3) - 5x> -3x - 6$
f. $-5x - 6xleq 8 - 8x - x$
you are playing a trivia game on your cell phone. you need a mean score of at least 90 points to advance to the next round. what scores in the fifth game will allow you to advance?
game 1: 95
game 2: 91
game 3: 77
game 4: 89
what if? you need a mean score of at least 85 points to advance to the next round. what scores in the fifth game will allow you to advance?

Explanation:

Step1: Recall mean formula

The mean of \(n\) numbers \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{x_1 + x_2+\cdots+x_n}{n}\). Here \(n = 5\), and let the score of the fifth - game be \(x\). The scores of the first four games are \(x_1=95\), \(x_2 = 91\), \(x_3=77\), \(x_4 = 89\).

Step2: Set up inequality for first case

We want \(\frac{95 + 91+77+89+x}{5}\geq90\). First, simplify the numerator of the left - hand side: \(95 + 91+77+89=352\). So the inequality becomes \(\frac{352 + x}{5}\geq90\).

Step3: Solve the inequality

Multiply both sides of the inequality \(\frac{352 + x}{5}\geq90\) by \(5\) to get \(352+x\geq450\). Then subtract \(352\) from both sides: \(x\geq450 - 352\), so \(x\geq98\).

Step4: Set up inequality for second case

We want \(\frac{95 + 91+77+89+x}{5}\geq85\). Since \(95 + 91+77+89 = 352\), the inequality is \(\frac{352+x}{5}\geq85\).

Step5: Solve the second inequality

Multiply both sides of \(\frac{352 + x}{5}\geq85\) by \(5\) to obtain \(352+x\geq425\). Subtract \(352\) from both sides: \(x\geq425 - 352\), so \(x\geq73\).

Answer:

To advance with a mean score of at least 90 points, the score in the fifth game should be \(x\geq98\). To advance with a mean score of at least 85 points, the score in the fifth game should be \(x\geq73\).