Problem 6 - Algebra Form and Function

Question

Are there values of \(x\) which satisfy the statements? Explain how you can tell without finding, or attempting to find, the values. $$ |3-x|-1<0 $$

Step-by-Step Solution

Answer

Answer: Yes, there are values of x that satisfy the given inequality.

Step 1: Set Up The Two Cases

We will split the inequality into two cases based on the possible values of the expression inside the absolute value: one for when \(3-x\) is non-negative, and one for when it is negative. Case 1: \(3-x\ge0\) or \(3\ge x\) Case 2: \(3-x<0\) or \(3

Step 2: Solve Each Case

Let's start with case 1. In this case, \(3-x\ge0\). Therefore, \(|3-x|=3-x\). Then the inequality becomes: $$ (3-x)-1<0 $$ Simplify and solve for \(x\): $$ 2-x<0\Rightarrow x>2 $$ Now, notice that \(x>2\) and \(3\ge x\) contradict each other. Since our solution contradicts the given condition for this case, there is no solution for the case 1. Next, let's consider case 2. In this case, \(3-x<0\). Therefore, \(|3-x|=-(3-x)\). Then the inequality becomes: $$ -(3-x)-1<0 $$ Simplify and solve for \(x\): $$ x-3-1<0\Rightarrow x<4 $$ This time, \(x<4\) and \(3

Key Concepts

Absolute Value

Absolute value refers to the distance a number is from zero on the number line, regardless of direction. This is why absolute value is always non-negative.
For a given real number x, the absolute value is denoted as \(|x|\). The absolute value function is defined as:

• \(|x| = x\) if \(x \ge 0\)
• \(|x| = -x\) if \(x < 0\)

Understanding this concept is vital when solving inequalities that involve absolute values, as it impacts how we set up and solve the equation or inequality.
For example, in the exercise, the expression \(|3-x|\) changes based on whether \(3-x\) is positive or negative.

Contradiction

A contradiction arises when a set of conditions is impossible to satisfy at the same time. In mathematical expressions, it's useful for showing when no solution exists.
In the exercise, if you have a situation where an inequality or equation gives a set of conditions that cannot both be true at the same time, it means the setup has a contradiction.
For instance, we found a contradiction in Case 1 of our original problem. Here, the inequality suggested \(x > 2\) but the given condition of Case 1 was \(3 \ge x\). Since \(x > 2\) and \(x \le 3\) overlap perfectly to describe a possible solution region, detecting where statements clash helps in understanding if there are no solutions or need alternative approaches.

Inequality Solving

Solving inequalities follows similar principles to solving equations, but with special attention to the direction of the inequality symbol. You want to isolate the variable on one side of the inequality.
Key steps include:

• Maintaining the direction of the inequality unless multiplying or dividing by a negative number, which reverses the direction.
• Simplifying both sides as much as possible.
• Interpreting the solution correctly in the context of the problem.

The primary goal in inequality solving is to figure out the range of values that satisfy the inequality. It often involves simplifying complex expressions and understanding the impact of operations such as addition, subtraction, multiplication, and division on the inequality's direction. In our exercise, solving the inequality \(|3-x|-1<0\) required careful execution of these steps across different cases.

Case Analysis

Case analysis is a powerful technique that involves breaking a problem into separate scenarios or cases to simplify the solving process. When facing an absolute value inequality, splitting into cases based on possible signs of the expressions inside provides clarity.
This approach involves identifying the different conditions that affect the problem.

• In our exercise, Case 1 assumed \(3-x\) was non-negative (\(3-x \ge 0\)), while Case 2 presumed \(3-x\) was negative (\(3-x < 0\)).
• Each case simplifies the inequality in a manner consistent with the signs being examined.

By solving for each scenario individually and then combining potential solutions, case analysis allows for logical deductions about the problem structure. This methodology ultimately helps identify contradictions or coherent solutions as seen in the original problem. It’s particularly useful when dealing with complex regions of number lines and range detections.

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Problem 6

Next

Problem 7

Other exercises in this chapter

Problem 5

Are there values of \(x\) which satisfy the statements? Explain how you can tell without finding, or attempting to find, the values. $$ |x-1|>2 $$

View solution

Problem 6

Write an inequality describing the given quantity. Example: An MP3 player can hold up to 120 songs. Solution: The number of songs is \(n\) where $0 \leq n \leq

View solution

Problem 7

Each of the inequalities can be solved by performing a single operation on both sides. State the operation, indicating whether or not the inequality changes dir

View solution

Problem 8

Solve the equations. $$ 0.2(g-6)=0.6(g-4) $$

View solution

Key Concepts

Absolute Value

Absolute value refers to the distance a number is from zero on the number line, regardless of direction. This is why absolute value is always non-negative.
For a given real number x, the absolute value is denoted as \(|x|\). The absolute value function is defined as:

• \(|x| = x\) if \(x \ge 0\)
• \(|x| = -x\) if \(x < 0\)

Understanding this concept is vital when solving inequalities that involve absolute values, as it impacts how we set up and solve the equation or inequality.
For example, in the exercise, the expression \(|3-x|\) changes based on whether \(3-x\) is positive or negative.

Contradiction

A contradiction arises when a set of conditions is impossible to satisfy at the same time. In mathematical expressions, it's useful for showing when no solution exists.
In the exercise, if you have a situation where an inequality or equation gives a set of conditions that cannot both be true at the same time, it means the setup has a contradiction.
For instance, we found a contradiction in Case 1 of our original problem. Here, the inequality suggested \(x > 2\) but the given condition of Case 1 was \(3 \ge x\). Since \(x > 2\) and \(x \le 3\) overlap perfectly to describe a possible solution region, detecting where statements clash helps in understanding if there are no solutions or need alternative approaches.

Inequality Solving

Solving inequalities follows similar principles to solving equations, but with special attention to the direction of the inequality symbol. You want to isolate the variable on one side of the inequality.
Key steps include:

• Maintaining the direction of the inequality unless multiplying or dividing by a negative number, which reverses the direction.
• Simplifying both sides as much as possible.
• Interpreting the solution correctly in the context of the problem.

The primary goal in inequality solving is to figure out the range of values that satisfy the inequality. It often involves simplifying complex expressions and understanding the impact of operations such as addition, subtraction, multiplication, and division on the inequality's direction. In our exercise, solving the inequality \(|3-x|-1<0\) required careful execution of these steps across different cases.

Case Analysis

Case analysis is a powerful technique that involves breaking a problem into separate scenarios or cases to simplify the solving process. When facing an absolute value inequality, splitting into cases based on possible signs of the expressions inside provides clarity.
This approach involves identifying the different conditions that affect the problem.

• In our exercise, Case 1 assumed \(3-x\) was non-negative (\(3-x \ge 0\)), while Case 2 presumed \(3-x\) was negative (\(3-x < 0\)).
• Each case simplifies the inequality in a manner consistent with the signs being examined.

By solving for each scenario individually and then combining potential solutions, case analysis allows for logical deductions about the problem structure. This methodology ultimately helps identify contradictions or coherent solutions as seen in the original problem. It’s particularly useful when dealing with complex regions of number lines and range detections.