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Zorluk: ZorSolving Linear Inequalities

A manufacturing company determines that its weekly profit from producing xx batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches xx must satisfy the inequality:

5(6−x)3−3(x+2)4>2x−11\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

Cevap: 4 batches

Cevap

The greatest number of whole batches the company can produce is 4.
Solving the inequality yields x<23453≈4.415x < \frac{234}{53} \approx 4.415. The greatest integer value that satisfies this condition is 4.

Adım Adım Çözüm

1
Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.
20(6−x)−9(x+2)>24x−13220(6 - x) - 9(x + 2) > 24x - 132
Multiplying by 12 eliminates the fractions, making the inequality easier to solve.
2
Distribute the constants on the left side of the inequality.
120−20x−9x−18>24x−132120 - 20x - 9x - 18 > 24x - 132
Distributing 20 to (6−x)(6 - x) yields 120−20x120 - 20x, and distributing −9-9 to (x+2)(x + 2) yields −9x−18-9x - 18.
3
Combine like terms on the left side of the inequality.
102−29x>24x−132102 - 29x > 24x - 132
Combining 120−18120 - 18 gives 102102, and combining −20x−9x-20x - 9x gives −29x-29x.
4
Subtract 24x24x from both sides to group the variable terms on the left side.
102−53x>−132102 - 53x > -132
This groups all terms containing the variable xx on one side of the inequality.
5
Subtract 102 from both sides to isolate the variable term.
−53x>−234-53x > -234
This isolates the term containing xx on the left side of the inequality.
6
Divide both sides by −53-53 and reverse the inequality sign.
x<23453x < \frac{234}{53}
Dividing by a negative number requires reversing the direction of the inequality sign.
7
Evaluate the fraction as a decimal and determine the greatest integer value of xx that satisfies the inequality.
x<4.415x < 4.415, which means the greatest integer is 4.
Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Anahtar Kavram

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternatif Yöntem

Instead of solving algebraically, you can test integer values for xx directly in the inequality. Testing x=4x = 4 gives 103−184=3.33−4.5=−1.17\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17, which is greater than 2(4)−11=−32(4) - 11 = -3 (True). Testing x=5x = 5 gives 53−214=1.67−5.25=−3.58\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58, which is not greater than 2(5)−11=−12(5) - 11 = -1 (False). This confirms 4 is the largest integer satisfying the inequality.
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Solving Linear Inequalities

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