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Chapter 5: Problem 43

a. Is the point (2,1) a solution to the inequality \(y<2 x+3 ?\) b. Is the point (2,1) a solution to the inequality \(x+y \leq 1 ?\) c. Is the point (2,1) a solution to the system of inequalities?

Short Answer

Expert verified
a. Yes, b. No, c. No

Step by step solution

01

Substitute point (2,1) into the first inequality

Substitute x = 2 and y = 1 into the inequality \(y < 2x + 3\). This gives \(1 < 2(2) + 3\).
02

Simplify the first inequality

Simplify the expression to check if the inequality holds: \(1 < 4 + 3 \), which simplifies to \(1 < 7\). Since this is true, the point (2,1) satisfies the first inequality.
03

Substitute point (2,1) into the second inequality

Substitute x = 2 and y = 1 into the inequality \(x + y \leq 1\). This gives \(2 + 1 \leq 1\).
04

Simplify the second inequality

Simplify the expression to check if the inequality holds: \(3 \leq 1\). Since this is false, the point (2,1) does not satisfy the second inequality.
05

Conclusion for the system of inequalities

Since the point (2,1) does not satisfy both inequalities simultaneously, it is not a solution to the system of inequalities.

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Key Concepts

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

system of inequalities
In algebra, a system of inequalities consists of multiple inequalities that are considered simultaneously. To determine if a certain point is a solution to this system, the point must satisfy all the inequalities in the system at once. For example, if a system of inequalities contains the read more...

In the original problem, we checked if the point (2,1) is a solution. Here's a quick breakdown:

  • First, we tested the inequality \( y < 2x + 3 \) and found it true for (2,1).
  • Next, we tested the inequality \( x + y \leq 1 \) and found it false for (2,1).
To be a solution to the system, (2,1) would need to work for both inequalities, but it didn't. Thus, it is not a solution for the system.
substitution method
The substitution method is a straightforward way to solve systems of equations or inequalities. It involves substituting values from one inequality or equation into another. Let's dive into how this method applied to our example.

  • First, we substituted the values \( x = 2 \) and \( y = 1 \) into the inequality \( y < 2x + 3 \).
  • This resulted in the expression \( 1 < 4 + 3 \), simplified as \( 1 < 7 \). Hence, (2,1) satisfied the first inequality.
  • Then, we substituted the same values into the inequality \( x + y \leq 1 \) to get \( 2 + 1 \leq 1 \), simplified as \( 3 \leq 1 \). Since this wasn't true, (2,1) didn't satisfy the second inequality.
By substituting values and simplifying, we can check if a point satisfies all conditions in a system.
algebraic inequalities
Algebraic inequalities express a relationship between expressions using inequality symbols like <, >, \leq, and \geq. These inequalities inform which values are greater or smaller relatively.

When working with algebraic inequalities:
  • Isolate the variable on one side.
  • Simplify the equation step-by-step.
  • Always check your solution by substituting back into the original inequality.
  • Remember that multiplying or dividing by a negative number flips the inequality symbol.
In our example, we used the substitution method to test if (2,1) fit in our inequalities: \( y < 2x + 3 \) and \( x + y \leq 1 \). We found it satisfied the first but not the second.

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Most popular questions from this chapter

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