Problem 58
Question
Will help you prepare for the material covered in the next section. Is \(2 x-3 y \geq 6\) a true statement for \(x=0\) and \(y=0 ?\)
Step-by-Step Solution
Verified Answer
No, it is not a true statement that \(2x - 3y ≥ 6\) for \(x = 0\) and \(y = 0\).
1Step 1: Substituting The Values
Substitute \(x = 0\) and \(y = 0\) into the inequality \(2x - 3y ≥ 6\). This yields \(2(0) - 3(0) ≥ 6\), which simplifies to \(0 ≥ 6\).
2Step 2: Analyze The Result
In this step, it is evident that 0 is not greater than or equal to 6.
Key Concepts
Solving InequalitiesSubstitution MethodAlgebraic Expressions
Solving Inequalities
When we talk about solving inequalities, we're exploring the process of finding the values of variables that make an inequality true. Unlike equations, inequalities indicate a range of possible solutions, rather than just one.
Consider an inequality like \(4x + 5 > 9\). To solve it, we do something very similar to solving equations: we isolate the variable. Subtract 5 from both sides, getting \(4x > 4\), and then divide by 4, which gives us \(x > 1\). This tells us that any number greater than 1 makes the inequality true.
Your goal with inequalities is to maintain their balance while isolating the variable. Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. This is crucial to keep in mind to avoid common mistakes and ensure that the final solution is correct.
Consider an inequality like \(4x + 5 > 9\). To solve it, we do something very similar to solving equations: we isolate the variable. Subtract 5 from both sides, getting \(4x > 4\), and then divide by 4, which gives us \(x > 1\). This tells us that any number greater than 1 makes the inequality true.
Your goal with inequalities is to maintain their balance while isolating the variable. Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. This is crucial to keep in mind to avoid common mistakes and ensure that the final solution is correct.
Substitution Method
The substitution method is a fundamental technique used in algebra to solve systems of equations, and it can also be applied to inequalities. This method involves replacing variables with their known values to simplify the problem and potentially reveal the solution.
For example, if we have two variables, \(x\) and \(y\), and we know that \(x = 2\), we can substitute 2 for every occurrence of \(x\) in our equation or inequality. So, if the inequality is \(3x + 4y > 12\) and we know \(x = 2\), it becomes \(3(2) + 4y > 12\), which simplifies to \(6 + 4y > 12\). We can then solve for \(y\) using the methods for solving inequalities.
Using substitution helps to clarify what you're working with and to directly test whether an inequality holds true for given values, as shown in the original exercise with \(x = 0\) and \(y = 0\).
For example, if we have two variables, \(x\) and \(y\), and we know that \(x = 2\), we can substitute 2 for every occurrence of \(x\) in our equation or inequality. So, if the inequality is \(3x + 4y > 12\) and we know \(x = 2\), it becomes \(3(2) + 4y > 12\), which simplifies to \(6 + 4y > 12\). We can then solve for \(y\) using the methods for solving inequalities.
Using substitution helps to clarify what you're working with and to directly test whether an inequality holds true for given values, as shown in the original exercise with \(x = 0\) and \(y = 0\).
Algebraic Expressions
At the heart of algebra are algebraic expressions, which consist of variables, numbers, and operations. Think of them as sentences in the language of mathematics, where numbers and variables are combined using addition, subtraction, multiplication, and division.
An example of an algebraic expression is \(3x^2 - 2x + 7\). This expression can be modified, simplified, or evaluated by substituting variables with numbers. Understanding how to work with these expressions is pivotal in algebra, as they form the basis of both equations and inequalities.
To simplify an algebraic expression, we combine like terms, which are terms that have the same variable raised to the same power. For instance, in the expression \(2x + 3x^2 - 4 + x\), we combine \(2x\) and \(x\) to get \(3x\), but we can't combine those with \(3x^2\) because it has a different power. Simplifying expressions makes it easier to solve or analyze them and is a skill that is fundamental to mastering algebra.
An example of an algebraic expression is \(3x^2 - 2x + 7\). This expression can be modified, simplified, or evaluated by substituting variables with numbers. Understanding how to work with these expressions is pivotal in algebra, as they form the basis of both equations and inequalities.
To simplify an algebraic expression, we combine like terms, which are terms that have the same variable raised to the same power. For instance, in the expression \(2x + 3x^2 - 4 + x\), we combine \(2x\) and \(x\) to get \(3x\), but we can't combine those with \(3x^2\) because it has a different power. Simplifying expressions makes it easier to solve or analyze them and is a skill that is fundamental to mastering algebra.
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