Solve The Inequality: ${ 10k - 8 \ \textless \ 32 } O R Or Or { -9k + 9 \ \textless \ -54 \}
Introduction
Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving two separate inequalities and then combining the results to find the solution set.
Inequality 1: 10k - 8 < 32
Step 1: Add 8 to Both Sides
To isolate the term with the variable, we need to get rid of the constant term on the left side of the inequality. We can do this by adding 8 to both sides of the inequality.
10k - 8 + 8 < 32 + 8
This simplifies to:
10k < 40
Step 2: Divide Both Sides by 10
Now that we have isolated the term with the variable, we need to get rid of the coefficient on the left side of the inequality. We can do this by dividing both sides of the inequality by 10.
\frac{10k}{10} < \frac{40}{10}
This simplifies to:
k < 4
Conclusion
The solution to the first inequality is k < 4.
Inequality 2: -9k + 9 < -54
Step 1: Add 9 to Both Sides
To isolate the term with the variable, we need to get rid of the constant term on the left side of the inequality. We can do this by adding 9 to both sides of the inequality.
-9k + 9 + 9 < -54 + 9
This simplifies to:
-9k + 18 < -45
Step 2: Subtract 18 from Both Sides
Now that we have isolated the term with the variable, we need to get rid of the constant term on the left side of the inequality. We can do this by subtracting 18 from both sides of the inequality.
-9k + 18 - 18 < -45 - 18
This simplifies to:
-9k < -63
Step 3: Divide Both Sides by -9
Now that we have isolated the term with the variable, we need to get rid of the coefficient on the left side of the inequality. We can do this by dividing both sides of the inequality by -9.
\frac{-9k}{-9} < \frac{-63}{-9}
This simplifies to:
k > \frac{7}{3}
Conclusion
The solution to the second inequality is k > 7/3.
Combining the Results
To find the solution set, we need to combine the results of both inequalities. Since the first inequality is k < 4 and the second inequality is k > 7/3, we can combine them by finding the intersection of the two solution sets.
The intersection of the two solution sets is:
\frac{7}{3} < k < 4
This means that the solution set is all values of k that are greater than 7/3 and less than 4.
Conclusion
Introduction
In our previous article, we solved two separate inequalities and then combined the results to find the solution set. In this article, we will answer some common questions that students often have when it comes to solving inequalities.
Q: What is the difference between an inequality and an equation?
A: An equation is a mathematical statement that says two expressions are equal, while an inequality is a mathematical statement that says one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.
Q: How do I solve an inequality with a variable on both sides?
A: To solve an inequality with a variable on both sides, you need to get all the variables on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality.
Q: What is the order of operations when solving an inequality?
A: The order of operations when solving an inequality is the same as when solving an equation: parentheses, exponents, multiplication and division, and addition and subtraction.
Q: Can I multiply or divide both sides of an inequality by a negative number?
A: No, you cannot multiply or divide both sides of an inequality by a negative number. This is because it would change the direction of the inequality sign.
Q: How do I solve an inequality with a fraction?
A: To solve an inequality with a fraction, you need to get rid of the fraction by multiplying both sides of the inequality by the denominator.
Q: What is the difference between a strict inequality and a non-strict inequality?
A: A strict inequality is an inequality that uses a strict inequality sign, such as < or >. A non-strict inequality is an inequality that uses a non-strict inequality sign, such as ≤ or ≥.
Q: Can I combine two inequalities with the same variable?
A: Yes, you can combine two inequalities with the same variable by finding the intersection of the two solution sets.
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution set. If the inequality is strict, you will use an open circle. If the inequality is non-strict, you will use a closed circle.
Q: What is the solution set of an inequality?
A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality.
Conclusion
In this article, we answered some common questions that students often have when it comes to solving inequalities. We covered topics such as the difference between an inequality and an equation, solving inequalities with variables on both sides, and graphing inequalities on a number line. We hope that this article has been helpful in clarifying any confusion you may have had about solving inequalities.
Additional Resources
Practice Problems
- Solve the inequality: 2x + 5 < 11
- Solve the inequality: x - 3 > 7
- Solve the inequality: 4x - 2 ≥ 10
- Solve the inequality: x + 2 ≤ 9
- Solve the inequality: 3x + 1 > 14
Answer Key
- x < 3
- x > 10
- x ≥ 3
- x ≤ 7
- x > 4