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Chapter 4: Problem 508

Determine whether each ordered pair is a solution to the inequality \(x+y>4\) : (a) (5,1) (b) (-2,6) c) (3,2) (d) (10,-5) (e) (0,0)

Short Answer

Expert verified
(a) Yes, (b) No, (c) Yes, (d) Yes, (e) No

Step by step solution

01

- Understand the inequality

The given inequality is 饾懃+饾懄>4. To determine if an ordered pair (饾懃,饾懄) is a solution, substitute 饾懃 and 饾懄 into the inequality and verify if the resulting statement is true.
02

- Test pair (5,1)

Substitute 饾懃 = 5 and 饾懄 = 1 into the inequality:5 + 1 > 46 > 4, which is true. Therefore, (5,1) is a solution.
03

- Test pair (-2,6)

Substitute 饾懃 = -2 and 饾懄 = 6 into the inequality:-2 + 6 > 44 = 4, which is not greater. Therefore, (-2,6) is not a solution.
04

- Test pair (3,2)

Substitute 饾懃 = 3 and 饾懄 = 2 into the inequality:3 + 2 > 45 > 4, which is true. Therefore, (3,2) is a solution.
05

- Test pair (10,-5)

Substitute 饾懃 = 10 and 饾懄 = -5 into the inequality:10 - 5 > 45 > 4, which is true. Therefore, (10,-5) is a solution.
06

- Test pair (0,0)

Substitute 饾懃 = 0 and 饾懄 = 0 into the inequality:0 + 0 > 40 > 4, which is false. Therefore, (0,0) is not a solution.

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Key Concepts

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

ordered pairs
In mathematics, an **ordered pair** is used to represent a point or location on a coordinate plane. It consists of two components: the first number (饾懃) indicates the horizontal position, while the second number (饾懄) indicates the vertical position. For example, (3, 2) tells us that the point is at an intersection where 饾懃 is 3 and 饾懄 is 2.

Ordered pairs are fundamental in solving inequalities that involve two variables. They allow us to graph points and understand relationships between variables. By substituting these values into an equation or inequality, we can verify if a point satisfies the given condition.
solution testing
Now, let's talk about **solution testing** for inequalities. When given an inequality like 饾懃 + 饾懄 > 4, we need to determine if a particular ordered pair satisfies the condition. This process involves substituting the 饾懃 and 饾懄 values from the ordered pair into the inequality.

Here are the steps for solution testing:
  • Take the ordered pair (饾懃, 饾懄).
  • Substitute 饾懃 and 饾懄 into the inequality.
  • Simplify the expression.
  • Check if the resulting statement is true.
If the statement is true, the ordered pair is a solution to the inequality. If it is false, the ordered pair is not a solution.
linear inequalities
A **linear inequality** is similar to a linear equation but instead of an equal sign, it has an inequality sign (>, <, 鈮�, 鈮�). In this case, 饾懃 + 饾懄 > 4 represents a linear inequality. The graph of a linear inequality like this divides the coordinate plane into two regions: one region where the inequality is true and another where it is false.

For the inequality 饾懃 + 饾懄 > 4, any point (饾懃, 饾懄) above the line formed by 饾懃 + 饾懄 = 4 will satisfy the inequality, making it a solution. Points on the line do not satisfy the inequality as the inequality is 'strict' (just >, not 鈮�). Understanding the geometric representation helps to visualize and solve problems more intuitively.
substitution method
The **substitution method** is a simple yet powerful tool used to test whether an ordered pair is a solution to an equation or inequality. In this method, you substitute the values of the ordered pair (饾懃, 饾懄) into the given expression.

For example, to check if the pair (5, 1) is a solution for 饾懃 + 饾懄 > 4:
  • Substitute 饾懃 = 5 and 饾懄 = 1 into the inequality.
  • Perform the addition: 5 + 1 = 6.
  • Check the resulting statement: 6 > 4. Since this is true, (5, 1) is a solution.
This method helps us systematically verify each pair quickly and accurately. With practice, it becomes second nature for solving inequalities and other algebraic problems.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-5,-3) and (4,-6)

Graph the linear inequality \(y>-2 x+1\)

Graph the linear inequality \(x+5 y<-5\)

Graph the linear inequality \(y \geq \frac{3}{2} x\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=-2 x+3\), point(2,2)
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