Problem 42 - Calculus Single Variable

Question

Sketch the set on a real number line. \(\left\\{t:(t-4)^{2}<(\mathrm{t}-2)^{2}\right.\) or \(\left.|t+1| \leq 4\right\\}\)

Step-by-Step Solution

Answer

The solution set is \([-5, \infty)\) on the number line, starting from -5 and extending infinitely to the right.

Step 1: Understand the Inequalities

We have two conditions to consider: 1. \((t-4)^2 < (t-2)^2\)2. \(|t+1| \leq 4\).We'll solve these inequalities separately and then find the union of their solutions.

Step 2: Solve the First Inequality

The inequality \((t-4)^2 < (t-2)^2\) can be expanded and simplified:1. Expand both sides: \((t-4)^2 = t^2 - 8t + 16\) and \((t-2)^2 = t^2 - 4t + 4\).2. Set up the inequality: \(t^2 - 8t + 16 < t^2 - 4t + 4\).3. Simplify by subtracting \(t^2\) from both sides: \(-8t + 16 < -4t + 4\).4. Rearrange terms: \(-8t + 4t < 4 - 16\).5. Simplify: \(-4t < -12\).6. Divide by \(-4\) (remember to reverse the inequality): \(t > 3\).

Step 3: Solve the Second Inequality

For the inequality \(|t+1| \leq 4\):1. This means \(-4 \leq t+1 \leq 4\).2. Solve for \(t\) by subtracting 1: - \(-4 - 1 \leq t \leq 4 - 1\).3. Simplify: \(-5 \leq t \leq 3\).

Step 4: Find the Union of Solutions

Now, combine the solutions from Steps 2 and 3:1. First inequality: \(t > 3\).2. Second inequality: \(-5 \leq t \leq 3\).The union of these two inequalities is the set of \(t\) values that satisfy **either** condition. Therefore, the solution is \(-5 \leq t \leq 3\) or \(t > 3\), which simplifies to: \(-5 \leq t < \infty\).

Step 5: Sketch the Solution on a Number Line

Draw a real number line and shade the region from \(-5\) to \(\infty\):- Use a closed circle at \(-5\), indicating \(-5\) is included.- Shade fully from \(-5\), extending to the right infinitely to represent \(t < \infty\).- Open the circle on 3 is not necessary since the solution includes \(-5 \leq t < \infty\).

Key Concepts

Absolute Value Inequality

In the world of inequalities, an absolute value inequality involves finding unknown values within a specific distance from a point on the number line.
Specifically, solving something like \(|t+1| \leq 4\) requires understanding what 'absolute value' truly signifies.

• Absolute value, \(|a|\), refers to the non-negative distance of a number \(a\) from zero on a number line.
• A statement like \(|t+1| \leq 4\) indicates that \(t + 1\) must lie within 4 units of zero.

Thus, these inequalities are split into two separate inequality expressions: one stating that \(t + 1\) is greater than or equal to -4, and the other that it is less than or equal to 4.
These can be written as the compound inequality: \-4 \leq t+1 \leq 4\.
By isolating \(t\), we subtract 1 from all parts resulting in \-5 \leq t \leq 3\.
This indicates that \(t\) falls within the range from -5 to 3 on a number line.

Quadratic Inequality

Quadratic inequalities are simply inequalities that involve squares of the variable, typically in the form of \( ax^2 + bx + c \).
In this case, we delve into the inequality \( (t-4)^2 < (t-2)^2 \).
The first step to tackling such inequalities is to expand both sides:

• \((t-4)^2\) becomes \(t^2 - 8t + 16\).
• Meanwhile, \((t-2)^2\) simplifies to \(t^2 - 4t + 4\).

The equation then simplifies by subtracting the common term of \(t^2\), which eventually boils down to solving a linear inequality: \ -4t < -12 \By dividing through by \(-4\), which importantly reverses the inequality sign, we find \(t > 3\).
This specific inequality solution indicates that \(t\) must be greater than three.

Union of Solutions

The preparation for a union of solutions combines all possible solutions that satisfy either one of the inequalities.
It involves merging separate pieces into one complete set based on the principle that solutions do not need to fit both conditions, but rather, at least one.

• From the absolute value inequality, we have the range \(-5 \leq t \leq 3\).
• From the quadratic inequality, \(t > 3\).

The union combines these results, highlighting that the solution encompasses all valid \(t\) values from both inequalities.
Graphically on a number line, this is expressed as \(-5 \leq t < \infty\).
This result shows inclusion of all numbers greater than or equal to -5, continuing without bound to infinity.
In this method, the union setup simplifies and ensures both ranges combine smoothly, justifying why \(-5 \leq t < \infty\) covers all possibilities logically and uniformly.