Solve The Inequality And Graph The Solution On The Line Provided.${2x - 12 \ \textless \ -10}$Answer:Attempt 1 Out Of
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Introduction
In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities and graphing the solution on a number line.
What is a Linear Inequality?
A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable. The inequality can be either greater than (<), less than (>, or equal to (≤). For example, 2x - 12 < -10 is a linear inequality.
Solving Linear Inequalities
To solve a linear inequality, we need to isolate the variable x. We can do this by adding or subtracting the same value to both sides of the inequality. We can also multiply or divide both sides of the inequality by the same non-zero value.
Step 1: Add or Subtract the Same Value
In the given inequality 2x - 12 < -10, we can add 12 to both sides to get 2x < 2.
Step 2: Divide Both Sides by the Same Non-Zero Value
Now, we can divide both sides of the inequality by 2 to get x < 1.
Graphing the Solution on a Number Line
Once we have solved the inequality, we can graph the solution on a number line. A number line is a line that represents all the real numbers. We can use a number line to visualize the solution to an inequality.
Step 1: Draw a Number Line
Draw a number line with a series of points marked at equal intervals.
Step 2: Shade the Solution
To graph the solution to the inequality x < 1, we need to shade all the points to the left of 1.
Example
Let's consider another example: 3x + 5 > 11.
Step 1: Subtract 5 from Both Sides
Subtract 5 from both sides of the inequality to get 3x > 6.
Step 2: Divide Both Sides by 3
Now, we can divide both sides of the inequality by 3 to get x > 2.
Step 3: Graph the Solution
To graph the solution to the inequality x > 2, we need to shade all the points to the right of 2.
Conclusion
In conclusion, solving linear inequalities and graphing the solution on a number line is an essential skill in mathematics. By following the steps outlined in this article, you can solve linear inequalities and visualize the solution on a number line.
Tips and Tricks
- When solving linear inequalities, always isolate the variable x.
- When graphing the solution on a number line, shade all the points that satisfy the inequality.
- Use a number line to visualize the solution to an inequality.
Frequently Asked Questions
- What is a linear inequality? A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable.
- How do I solve a linear inequality? To solve a linear inequality, you need to isolate the variable x by adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides of the inequality by the same non-zero value.
- How do I graph the solution to a linear inequality on a number line? To graph the solution to a linear inequality on a number line, you need to shade all the points that satisfy the inequality.
References
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Introduction
In our previous article, we discussed how to solve linear inequalities and graph the solution on a number line. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities.
Q&A
Q: What is a linear inequality?
A: A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, you need to isolate the variable x by adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides of the inequality 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 ax + b < c, where a, b, and c are constants, and x is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form of ax^2 + bx + c < d, where a, b, c, and d are constants, and x is the variable.
Q: How do I graph the solution to a linear inequality on a number line?
A: To graph the solution to a linear inequality on a number line, you need to shade all the points that satisfy the inequality.
Q: What is the concept of "less than" and "greater than" in inequalities?
A: In inequalities, "less than" (<) means that the value on the left-hand side is smaller than the value on the right-hand side. "Greater than" (>) means that the value on the left-hand side is larger than the value on the right-hand side.
Q: Can I use the same steps to solve a quadratic inequality as I would a linear inequality?
A: No, you cannot use the same steps to solve a quadratic inequality as you would a linear inequality. Quadratic inequalities require a different set of steps to solve.
Q: How do I determine the direction of the inequality when solving a quadratic inequality?
A: When solving a quadratic inequality, you need to determine the direction of the inequality by considering the sign of the coefficient of the x^2 term.
Q: Can I use a number line to visualize the solution to a quadratic inequality?
A: Yes, you can use a number line to visualize the solution to a quadratic inequality.
Tips and Tricks
- When solving linear inequalities, always isolate the variable x.
- When graphing the solution to a linear inequality on a number line, shade all the points that satisfy the inequality.
- Use a number line to visualize the solution to an inequality.
- When solving quadratic inequalities, consider the sign of the coefficient of the x^2 term to determine the direction of the inequality.
Frequently Asked Questions
- What is a linear inequality? A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable.
- How do I solve a linear inequality? To solve a linear inequality, you need to isolate the variable x by adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides of the inequality by the same non-zero value.
- What is the difference between a linear inequality and a quadratic inequality? A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form of ax^2 + bx + c < d, where a, b, c, and d are constants, and x is the variable.