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Chapter 5: Problem 16

Determine whether the ordered pair is a solution to the inequality. \(y \geq(x-3)^{2}\) a. (-3,30) b. (1,4) c. (5,5)

Short Answer

Expert verified
The ordered pairs that are solutions to the inequality are (1, 4) and (5, 5).

Step by step solution

01

Identify the inequality

The given inequality is The inequality to determine whether the ordered pairs are solutions for is:
02

Check the first ordered pair (-3, 30)

Substitute y = 30 and x = -3 into the inequality: 30 30 36 Since 30 is not greater than or equal to 36 , the first ordered pair is not a solution.
03

Check the second ordered pair (1, 4)

Substitute y = 4 and x = 1 into the inequality: 4 4 0 Since 4 is greater than or equal to 0 , the second ordered pair is a solution.
04

Check the third ordered pair (5, 5)

Substitute y = 5 and x = 5 into the inequality: 5 5 4 Since 5 is greater than or equal to 4 , the third ordered pair is a solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
An ordered pair consists of two elements arranged in a specific order. It's typically written as \((x, y)\). The first element is the x-coordinate and the second element is the y-coordinate.
For example, in the ordered pair \((-3, 30)\), -3 is the x-coordinate, and 30 is the y-coordinate.
Ordered pairs are often used to represent points in the coordinate plane. By substituting the x and y values of an ordered pair into an inequality or equation, we can determine whether that point satisfies the conditions.
In this exercise, we will check if the given points satisfy the inequality \((y \geq (x-3)^{2})\).
Inequalities
An inequality is a mathematical statement that relates expressions that are not necessarily equal. It uses symbols such as \(<\), \(>\), \(\leq\), and \(\geq\).
In this problem, we are given the inequality \((y \geq (x-3)^{2})\). This means that for a pair \((x, y)\) to be a solution, the value of y must be greater than or equal to \((x-3)^{2})\).
To check if an ordered pair satisfies this inequality, substitute the x and y values from the pair into the inequality and see if the inequality holds true.
Substitution Method
The substitution method involves replacing variables in an equation or inequality with given values to determine if the statement holds true.
Let's go through each ordered pair using the substitution method:
  • For \((-3, 30)\):

    • Substitute \(x = -3\) and \(y = 30\) into the inequality:
    \((30 \geq (-3-3)^{2})\)
    \(30 \geq 36\)
    Since 30 is not greater than or equal to 36, \((-3, 30)\) is not a solution.
  • For \((1, 4)\):

    • Substitute \(x = 1\) and \(y = 4\) into the inequality:
    \((4 \geq (1-3)^{2})\)
    \(4 \geq 4\)
    Since 4 is equal to 4, \((1, 4)\) is a solution.
  • For \((5, 5)\):

    • Substitute \(x = 5\) and \(y = 5\) into the inequality:
    \((5 \geq (5-3)^{2})\)
    \((5 \geq 4)\)
    Since 5 is greater than 4, \((5, 5)\) is a solution.
Quadratic Equations
A quadratic equation is a second-order polynomial equation in a single variable x, with a non-zero coefficient for \(x^{2}\). It has the standard form \((ax^{2} + bx + c = 0)\).
In our inequality \((y \geq (x-3)^{2})\), the term \((x-3)^{2}\) is a quadratic expression. When solving inequalities involving quadratic expressions, the y-value must be compared to the quadratic value.
In our example, we substituted the values of x from the ordered pairs into the quadratic expression \((x-3)^{2}\), solved it, and then compared the resulting value to y.
This process helps us determine if the quadratic value satisfies the inequality condition for y.

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Most popular questions from this chapter

Use the given constraints to find the maximum value of the objective function and the ordered pair \((x, y)\) that produces the maximum value. \(x \geq 0, y \geq 0\) \(x+y \leq 20\) \(x+2 y \leq 36\) \(x \leq 14\) a. Maximize: \(z=12 x+15 y\) b. Maximize: \(z=15 x+12 y\)

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} -4 x-8 y=2 \\ 2 x=8-4 y \end{array} $$

A college theater has a seating capacity of 2000 . It reserves \(x\) tickets for students and \(y\) tickets for general admission. For parts (a)-(d) write an inequality to represent the given statement. a. The total number of seats available is at most 2000 . b. The college wants to reserve at least 3 times as many student tickets as general admission tickets. c. The number of student tickets cannot be negative. d. The number of general admission tickets cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=\frac{3}{2} x-5\) \(6 x-4 y=20\) a. (2,-2) b. (-4,-11)
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