Examkin

Soru

Zorluk: ZorSolving Linear Inequalities

A manufacturing company produces two types of metal alloys. The production cost, in dollars per ton, of Alloy A is modeled by the expression 800−15(x−5)800 - 15(x - 5), where xx is the amount of stabilizer added in kilograms. The production cost of Alloy B, in dollars per ton, is modeled by the expression 520+5(x+3)520 + 5(x + 3). If the company requires the production cost of Alloy A to be strictly less than the production cost of Alloy B, what is the minimum integer amount of stabilizer xx, in kilograms, that must be added?

Cevap: 18 kg

Cevap

The minimum integer amount of stabilizer that must be added is 1818 kg.
The inequality representing the condition is 800−15(x−5)<520+5(x+3)800 - 15(x - 5) < 520 + 5(x + 3). Expanding both sides gives 875−15x<535+5x875 - 15x < 535 + 5x. Isolating xx yields −20x<−340-20x < -340. Dividing by −20-20 and flipping the inequality sign results in x>17x > 17. The minimum integer value that is strictly greater than 1717 is 1818.

Adım Adım Çözüm

1
Set up the linear inequality using the cost expressions for Alloy A and Alloy B.
800−15(x−5)<520+5(x+3)800 - 15(x - 5) < 520 + 5(x + 3)
The cost of Alloy A must be strictly less than the cost of Alloy B.
2
Distribute and combine like terms to simplify both sides of the inequality.
875−15x<535+5x875 - 15x < 535 + 5x
Simplifying the expressions makes it easier to isolate the variable.
3
Isolate the variable term on one side of the inequality.
−20x<−340-20x < -340
Subtracting 5x5x and 875875 from both sides moves all variable terms to the left and constant terms to the right.
4
Divide both sides by the negative coefficient −20-20 and reverse the inequality sign.
x>17x > 17
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.
5
Identify the smallest integer that satisfies the inequality.
1818
The solution requires a strict inequality x>17x > 17, so the smallest integer value that is strictly greater than 1717 is 1818.

Anahtar Kavram

Solving multi-step linear inequalities with variables on both sides, including reversing the inequality sign when dividing by a negative number.
Bu soruyu puanla

Konular

Solving Linear Inequalities

Bu konudan daha fazla soru