Complete The Steps To Solve The Inequality:$0.2(x+20)-3\ \textgreater \ -7-6.2x$1. Use The Distributive Property: ${ 0.2x + 4 - 3 \ \textgreater \ -7 - 6.2x }$2. Combine Like Terms: $[ 0.2x + 1 \ \textgreater \
Introduction
Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will guide you through the steps to solve the inequality .
Step 1: Use the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to expand expressions involving parentheses. In this case, we need to apply the distributive property to the left-hand side of the inequality.
{ 0.2(x+20)-3 > -7-6.2x \}
Using the distributive property, we can rewrite the expression as:
{ 0.2x + 4 - 3 > -7 - 6.2x \}
Step 2: Combine Like Terms
Now that we have applied the distributive property, we can combine like terms on both sides of the inequality. Like terms are expressions that contain the same variable(s) raised to the same power.
{ 0.2x + 4 - 3 > -7 - 6.2x \}
Combining like terms, we get:
{ 0.2x + 1 > -7 - 6.2x \}
Step 3: Add 6.2x to Both Sides
To isolate the variable on one side of the inequality, we need to add to both sides of the inequality. This will allow us to combine like terms and simplify the expression.
{ 0.2x + 1 > -7 - 6.2x \}
Adding to both sides, we get:
{ 7.2x + 1 > -7 \}
Step 4: Subtract 1 from Both Sides
Next, we need to subtract 1 from both sides of the inequality to isolate the term containing the variable .
{ 7.2x + 1 > -7 \}
Subtracting 1 from both sides, we get:
{ 7.2x > -8 \}
Step 5: Divide Both Sides by 7.2
Finally, we need to divide both sides of the inequality by 7.2 to solve for .
{ 7.2x > -8 \}
Dividing both sides by 7.2, we get:
{ x > -\frac{8}{7.2} \}
Simplifying the Expression
To simplify the expression, we can divide the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 8 and 7.2 is 0.4.
{ x > -\frac{8}{7.2} \}
Simplifying the expression, we get:
{ x > -\frac{20}{18} \}
Conclusion
In this article, we have guided you through the steps to solve the inequality . We have applied the distributive property, combined like terms, added and subtracted terms, and finally divided both sides by 7.2 to solve for . The final solution is .
Discussion
Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts such as the distributive property, combining like terms, and isolating variables. In this article, we have provided a step-by-step guide to solving inequalities, and we hope that it will be helpful to students and educators alike.
Additional Resources
For more information on solving inequalities, we recommend the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Wolfram Alpha: Solving Inequalities
FAQs
Q: What is the distributive property? A: The distributive property is a fundamental concept in algebra that allows us to expand expressions involving parentheses.
Q: How do I combine like terms? A: To combine like terms, we need to add or subtract expressions that contain the same variable(s) raised to the same power.
Q: How do I isolate a variable on one side of an inequality? A: To isolate a variable on one side of an inequality, we need to add or subtract terms to both sides of the inequality, while maintaining the direction of the inequality.
Q: What is an inequality?
A: An inequality is a mathematical expression that compares two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to expand expressions involving parentheses. It states that for any numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
Q: How do I combine like terms?
A: To combine like terms, we need to add or subtract expressions that contain the same variable(s) raised to the same power. For example, if we have the expression 2x + 3x, we can combine the like terms by adding them together:
2x + 3x = 5x
Q: How do I isolate a variable on one side of an inequality?
A: To isolate a variable on one side of an inequality, we need to add or subtract terms to both sides of the inequality, while maintaining the direction of the inequality. For example, if we have the inequality 2x + 3 > 5, we can isolate the variable x by subtracting 3 from both sides:
2x + 3 - 3 > 5 - 3 2x > 2
Q: How do I divide both sides of an inequality by a number?
A: To divide both sides of an inequality by a number, we need to multiply both sides of the inequality by the reciprocal of the number. For example, if we have the inequality 4x > 12, we can divide both sides by 4 by multiplying both sides by 1/4:
(4x) / 4 > 12 / 4 x > 3
Q: What is the difference between an inequality and an equation?
A: An inequality is a mathematical expression that compares two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. An equation, on the other hand, is a mathematical expression that states that two values are equal. For example, the expression 2x + 3 = 5 is an equation, while the expression 2x + 3 > 5 is an inequality.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, we need to isolate the variable on one side of the inequality, while maintaining the direction of the inequality. We can do this by adding or subtracting terms to both sides of the inequality, and then dividing both sides by a number if necessary.
Q: What is the solution to a linear inequality?
A: The solution to a linear inequality is the set of all values that satisfy the inequality. For example, if we have the inequality x > 2, the solution is all values greater than 2, including 2.5, 3, 3.5, and so on.
Q: How do I graph a linear inequality?
A: To graph a linear inequality, we need to graph the corresponding linear equation and then shade the region that satisfies the inequality. For example, if we have the inequality x > 2, we can graph the line x = 2 and then shade the region to the right of the line.
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 ax + b > c, where a, b, and c are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax^2 + bx + c > 0, where a, b, and c are constants.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, we need to factor the quadratic expression and then use the sign of the expression to determine the solution. We can also use the quadratic formula to solve the inequality.
Q: What is the solution to a quadratic inequality?
A: The solution to a quadratic inequality is the set of all values that satisfy the inequality. For example, if we have the inequality x^2 + 4x + 4 > 0, the solution is all values greater than -2.
Conclusion
In this article, we have answered some of the most frequently asked questions about solving inequalities. We have covered topics such as the distributive property, combining like terms, isolating variables, and graphing linear inequalities. We hope that this article has been helpful to you in your studies.