QUESTION 22.1 Given: \[$4 - 2x \ \textless \ 16\$\] Where \[$x \in \mathbb{R}\$\]2.1.1 Solve The Inequality. (2)2.1.2 Hence, Represent Your Answer To QUESTION 2.1.1 On A Number Line. (1)2.2 Solve Simultaneously For \[$x\$\]

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Introduction

In this article, we will explore the solution to a given inequality and represent the answer on a number line. We will also solve a simultaneous equation to find the value of the variable. The given inequality is 4−2x<164 - 2x < 16, where xx is a real number.

Solving the Inequality

To solve the inequality, we need to isolate the variable xx. We can start by subtracting 4 from both sides of the inequality:

4−2x−4<16−44 - 2x - 4 < 16 - 4

This simplifies to:

−2x<12-2x < 12

Next, we can divide both sides of the inequality by -2. However, when we divide by a negative number, the direction of the inequality sign changes:

−2x−2>12−2\frac{-2x}{-2} > \frac{12}{-2}

This simplifies to:

x>−6x > -6

Therefore, the solution to the inequality is x>−6x > -6.

Representing the Answer on a Number Line

To represent the answer on a number line, we need to plot the point -6 and shade the region to the right of it. This represents all the values of xx that are greater than -6.

Solving Simultaneously

Now, let's solve the simultaneous equation:

x>−6x > -6

x+2>4x + 2 > 4

We can start by subtracting 2 from both sides of the second inequality:

x>4−2x > 4 - 2

This simplifies to:

x>2x > 2

Therefore, the solution to the simultaneous equation is x>2x > 2.

Conclusion

In this article, we solved the inequality 4−2x<164 - 2x < 16 and represented the answer on a number line. We also solved a simultaneous equation to find the value of the variable. The solution to the inequality is x>−6x > -6, and the solution to the simultaneous equation is x>2x > 2.

Step-by-Step Solution

Step 1: Solve the Inequality

To solve the inequality, we need to isolate the variable xx. We can start by subtracting 4 from both sides of the inequality:

4−2x−4<16−44 - 2x - 4 < 16 - 4

This simplifies to:

−2x<12-2x < 12

Next, we can divide both sides of the inequality by -2. However, when we divide by a negative number, the direction of the inequality sign changes:

−2x−2>12−2\frac{-2x}{-2} > \frac{12}{-2}

This simplifies to:

x>−6x > -6

Step 2: Represent the Answer on a Number Line

To represent the answer on a number line, we need to plot the point -6 and shade the region to the right of it. This represents all the values of xx that are greater than -6.

Step 3: Solve Simultaneously

Now, let's solve the simultaneous equation:

x>−6x > -6

x+2>4x + 2 > 4

We can start by subtracting 2 from both sides of the second inequality:

x>4−2x > 4 - 2

This simplifies to:

x>2x > 2

Step 4: Conclusion

In this article, we solved the inequality 4−2x<164 - 2x < 16 and represented the answer on a number line. We also solved a simultaneous equation to find the value of the variable. The solution to the inequality is x>−6x > -6, and the solution to the simultaneous equation is x>2x > 2.

Key Takeaways

  • To solve an inequality, we need to isolate the variable.
  • When we divide by a negative number, the direction of the inequality sign changes.
  • To represent the answer on a number line, we need to plot the point and shade the region to the right of it.
  • To solve a simultaneous equation, we need to find the intersection of the two inequalities.

Frequently Asked Questions

Q: How do I solve an inequality?

A: To solve an inequality, we need to isolate the variable. We can start by subtracting or adding the same value to both sides of the inequality.

Q: What happens when we divide by a negative number?

A: When we divide by a negative number, the direction of the inequality sign changes.

Q: How do I represent the answer on a number line?

A: To represent the answer on a number line, we need to plot the point and shade the region to the right of it.

Q: How do I solve a simultaneous equation?

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical symbol such as <, >, ≤, or ≥. It is used to describe a relationship between two values or expressions.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable. You can start by subtracting or adding the same value to both sides of the inequality. You can also multiply or divide both sides by the same value, but be careful not to change the direction of the inequality sign.

Q: What happens when I divide by a negative number?

A: When you divide by a negative number, the direction of the inequality sign changes. For example, if you have the inequality x > 5 and you divide both sides by -2, the inequality becomes x < -2.5.

Q: How do I represent the answer on a number line?

A: To represent the answer on a number line, you need to plot the point and shade the region to the right of it. For example, if you have the inequality x > 5, you would plot the point 5 and shade the region to the right of it.

Q: What is a simultaneous equation?

A: A simultaneous equation is a system of two or more equations that are solved together to find the value of the variable. It is also known as a system of linear equations.

Q: How do I solve a simultaneous equation?

A: To solve a simultaneous equation, you need to find the intersection of the two equations. You can do this by adding or subtracting the two equations to eliminate one of the variables.

Q: What is the difference between an inequality and an equation?

A: An inequality is a statement that compares two expressions using a mathematical symbol such as <, >, ≤, or ≥. An equation is a statement that says two expressions are equal. For example, the inequality x > 5 is different from the equation x = 5.

Q: Can I use the same method to solve an inequality and an equation?

A: No, you cannot use the same method to solve an inequality and an equation. Inequalities require a different approach than equations.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

  • Changing the direction of the inequality sign when multiplying or dividing by a negative number
  • Not isolating the variable
  • Not checking the solution in the original inequality

Q: How do I check my solution in the original inequality?

A: To check your solution in the original inequality, you need to plug the value back into the inequality and see if it is true. If it is true, then the solution is correct.

Q: What are some real-life applications of inequalities?

A: Inequalities have many real-life applications, including:

  • Finance: Inequalities are used to compare the value of investments and determine the best option.
  • Science: Inequalities are used to describe the relationship between variables in scientific experiments.
  • Engineering: Inequalities are used to design and optimize systems.

Q: Can I use inequalities to solve problems in other areas of mathematics?

A: Yes, inequalities can be used to solve problems in other areas of mathematics, including algebra, geometry, and calculus.