What Is The First Step In Solving The Inequality $2x + 3 \geq 17$?A. Divide The Left Side By 2.B. Subtract 3 From Both Sides.C. Change The Direction Of The Inequality.D. Change The Inequality To $\ \textgreater \ $.
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Understanding the Basics of Inequalities
Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. Inequalities are denoted by the use of symbols such as <, >, ≤, ≥, and ≠. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation.
The First Step in Solving Inequalities
When solving an inequality, the first step is to isolate the variable on one side of the inequality. This is done by performing operations on both sides of the inequality to get the variable by itself. In the case of the inequality $2x + 3 \geq 17$, we need to isolate the variable x.
Analyzing the Options
Let's analyze the options given to determine the correct first step in solving the inequality.
Option A: Divide the left side by 2
Dividing the left side of the inequality by 2 would result in $x + \frac{3}{2} \geq \frac{17}{2}$. However, this is not the correct first step in solving the inequality. Dividing or multiplying both sides of an inequality by a constant can change the direction of the inequality, which is not the case here.
Option B: Subtract 3 from both sides
Subtracting 3 from both sides of the inequality would result in $2x \geq 14$. This is a step in the right direction, but it is not the first step in solving the inequality. We need to isolate the variable x, and subtracting 3 from both sides does not achieve this.
Option C: Change the direction of the inequality
Changing the direction of the inequality would result in $2x + 3 \leq 17$. This is not the correct first step in solving the inequality. Changing the direction of the inequality is not necessary in this case.
Option D: Change the inequality to $\ \textgreater \ $
Changing the inequality to $\ \textgreater \ $ would result in $2x + 3 > 17$. This is not the correct first step in solving the inequality. We need to isolate the variable x, and changing the inequality to $\ \textgreater \ $ does not achieve this.
The Correct First Step
The correct first step in solving the inequality $2x + 3 \geq 17$ is to subtract 3 from both sides. This results in $2x \geq 14$. This is the first step in isolating the variable x.
Why Subtracting 3 from Both Sides is the Correct First Step
Subtracting 3 from both sides of the inequality is the correct first step because it eliminates the constant term on the left side of the inequality. This allows us to isolate the variable x, which is the goal of solving the inequality.
The Importance of Isolating the Variable
Isolating the variable x is crucial in solving inequalities. By isolating the variable, we can determine the value or values of x that satisfy the inequality. In the case of the inequality $2x \geq 14$, we can divide both sides by 2 to get $x \geq 7$. This tells us that x is greater than or equal to 7.
Conclusion
In conclusion, the first step in solving the inequality $2x + 3 \geq 17$ is to subtract 3 from both sides. This results in $2x \geq 14$, which is the first step in isolating the variable x. By isolating the variable, we can determine the value or values of x that satisfy the inequality.
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Understanding the Basics of Inequalities
Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. Inequalities are denoted by the use of symbols such as <, >, ≤, ≥, and ≠. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation.
The First Step in Solving Inequalities
When solving an inequality, the first step is to isolate the variable on one side of the inequality. This is done by performing operations on both sides of the inequality to get the variable by itself. In the case of the inequality $2x + 3 \geq 17$, we need to isolate the variable x.
Analyzing the Options
Let's analyze the options given to determine the correct first step in solving the inequality.
Option A: Divide the left side by 2
Dividing the left side of the inequality by 2 would result in $x + \frac{3}{2} \geq \frac{17}{2}$. However, this is not the correct first step in solving the inequality. Dividing or multiplying both sides of an inequality by a constant can change the direction of the inequality, which is not the case here.
Option B: Subtract 3 from both sides
Subtracting 3 from both sides of the inequality would result in $2x \geq 14$. This is a step in the right direction, but it is not the first step in solving the inequality. We need to isolate the variable x, and subtracting 3 from both sides does not achieve this.
Option C: Change the direction of the inequality
Changing the direction of the inequality would result in $2x + 3 \leq 17$. This is not the correct first step in solving the inequality. Changing the direction of the inequality is not necessary in this case.
Option D: Change the inequality to $\ \textgreater \ $
Changing the inequality to $\ \textgreater \ $ would result in $2x + 3 > 17$. This is not the correct first step in solving the inequality. We need to isolate the variable x, and changing the inequality to $\ \textgreater \ $ does not achieve this.
The Correct First Step
The correct first step in solving the inequality $2x + 3 \geq 17$ is to subtract 3 from both sides. This results in $2x \geq 14$. This is the first step in isolating the variable x.
Why Subtracting 3 from Both Sides is the Correct First Step
Subtracting 3 from both sides of the inequality is the correct first step because it eliminates the constant term on the left side of the inequality. This allows us to isolate the variable x, which is the goal of solving the inequality.
The Importance of Isolating the Variable
Isolating the variable x is crucial in solving inequalities. By isolating the variable, we can determine the value or values of x that satisfy the inequality. In the case of the inequality $2x \geq 14$, we can divide both sides by 2 to get $x \geq 7$. This tells us that x is greater than or equal to 7.
Q&A: Solving Inequalities
Q: What is the first step in solving an inequality?
A: The first step in solving an inequality is to isolate the variable on one side of the inequality.
Q: How do I isolate the variable in an inequality?
A: To isolate the variable, you need to perform operations on both sides of the inequality to get the variable by itself.
Q: What operations can I perform on both sides of an inequality?
A: You can add or subtract the same value from both sides of an inequality, or multiply or divide both sides by the same non-zero value.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form of a linear equation, while a quadratic inequality is an inequality that can be written in the form of a quadratic equation.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you need to factor the quadratic expression, set each factor equal to zero, and solve for the variable.
Q: What is the importance of isolating the variable in an inequality?
A: Isolating the variable is crucial in solving inequalities because it allows you to determine the value or values of the variable that satisfy the inequality.
Q: Can I change the direction of an inequality when solving it?
A: Yes, you can change the direction of an inequality when solving it, but only if you multiply or divide both sides of the inequality by a negative value.
Q: What is the final step in solving an inequality?
A: The final step in solving an inequality is to check your solution by plugging it back into the original inequality.
Conclusion
In conclusion, solving inequalities requires a step-by-step approach. The first step is to isolate the variable on one side of the inequality, and the final step is to check your solution by plugging it back into the original inequality. By following these steps, you can solve any linear inequality and determine the value or values of the variable that satisfy the inequality.