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Difficulty: MediumLinear Inequalities in One Variable

To calculate the remaining budget, BB, in dollars, for a community project, a coordinator uses the formula B=1,200−3(2x+50)B = 1,200 - 3(2x + 50), where xx is the number of volunteer shifts scheduled. If the coordinator wants the remaining budget to be at most 450,whichofthefollowinginequalitiesrepresentsallpossiblevaluesof450, which of the following inequalities represents all possible values of x$?

  1. A
    x≤100x \le 100
  2. B
    x≥150x \ge 150
  3. C
    x≥10x \ge 10
  4. x≥100x \ge 100Answer

Answer

The correct answer is x≥100x \ge 100.
The correct answer is the inequality stating that xx is greater than or equal to 100100. Distributing −3-3 to both terms inside the parentheses gives 1,200−6x−150≤4501,200 - 6x - 150 \le 450. Combining the constant terms on the left side results in 1,050−6x≤4501,050 - 6x \le 450. Subtracting 1,0501,050 from both sides gives −6x≤−600-6x \le -600. Dividing both sides by −6-6 and reversing the inequality sign because of the division by a negative number yields x≥100x \ge 100.

Step-by-Step Solution

1
Substitute the formula for the remaining budget BB into the inequality B≤450B \le 450.
1,200−3(2x+50)≤4501,200 - 3(2x + 50) \le 450
To find the possible values of xx when the budget is at most 450450.
2
Distribute −3-3 to both terms inside the parentheses.
1,200−6x−150≤4501,200 - 6x - 150 \le 450
To simplify the expression by removing the parentheses.
3
Combine the constant terms 1,2001,200 and −150-150 on the left side.
1,050−6x≤4501,050 - 6x \le 450
To simplify the inequality further before isolating the variable.
4
Subtract 1,0501,050 from both sides of the inequality.
−6x≤−600-6x \le -600
To isolate the variable term on one side of the inequality.
5
Divide both sides of the inequality by −6-6 and reverse the inequality sign.
x≥100x \ge 100
Dividing by a negative number requires reversing the direction of the inequality sign to maintain the truth value.

Key Concept

Solving linear inequalities in one variable, including the distributive property and sign reversal when dividing by a negative number.
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Topics

Linear Inequalities in One Variable

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