Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![### Solve the Two-Step Inequalities
**Instructions:** Drag and drop the missing values in each two-step inequality solved below.
### Inequality Problems
**1.**
\[ -9x + 3 \leq 48 \]
\[ -9x \leq \ \_\_\ \]
\[ x \ \ \ \ \_\_ \]
**2.**
\[ \frac{n}{2} - 11 < -7 \]
\[ \frac{n}{2} < \ \_\_\ \]
\[ n \ \_\_ \]
**3.**
\[ 4b - 10 \geq -8 \]
\[ 4b \geq \ \_\_\ \]
\[ b \ \_\_ \]
**4.**
\[ \frac{1}{3}m + 11 > 41 \]
\[ \frac{1}{3}m > \ \_\_\ \]
\[ m \ \_\_ \]
**5.**
\[ -1.5c - 30 > -30 \]
\[ -1.5c > \ \_\_\ \]
\[ c \ \_\_ \]
**6.**
\[ \frac{p}{-6} + 28 \geq 30 \]
\[ \frac{p}{-6} \geq \ \_\_\ \]
\[ p \ \_\_ \]
### Drag and Drop Values
**Available values to drag** (located at the bottom of the image with arrows pointing to them):
- 30
- 4
- \< (less than)
- 90
- \>= (greater than or equal to)
- 8
- 0.5
- -5
- 45
- 2
- \<= (less than or equal to)
- -40
- -12
- 60
- **Note**: Each section of the inequality has two input boxes to be filled in with the correct values.
---
### Detailed Explanation of Diagrams:
The image above contains six two-step inequality problems that require the user to fill in the missing solutions and inequalities. Below these problems, there's a series of numerical and inequality symbols within a draggable interface that the user can utilize to solve each part of the problems. Drag the correct value or symbol into place to complete the inequality transformations accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53fa1db3-32f8-4c94-8cb8-72dc1b15d364%2F920e2dbc-39d5-4bde-a912-83d937d55403%2F9ntzb9l_processed.png&w=3840&q=75)
![### Inequality Matching Exercise
#### Objective:
Solve each inequality and drag the orange piece with the matching solution next to the inequality. Note: Not all orange pieces will be used.
---
#### Given Inequalities:
1. **First Inequality:**
\[
-4d + 28 \leq 10
\]
2. **Second Inequality:**
\[
\frac{d}{3.5} - 13 > -17
\]
3. **Third Inequality:**
\[
\frac{2}{3}d - 28 < 2
\]
4. **Fourth Inequality:**
\[
\frac{d}{-2} + 9 \geq -49
\]
---
#### Possible Solutions:
1. \( d < 20 \)
2. \( d < -7.5 \)
3. \( d < -14 \)
4. \( d \geq -80 \)
5. \( d \geq 4.5 \)
6. \( d < -80 \)
7. \( d < 45 \)
---
#### Activity Instructions:
For each inequality, solve for \( d \) and identify which of the provided solutions matches. Drag the corresponding orange piece with the matching solution next to the relevant inequality.
---
**Graphical Explanation:**
- **Visual Arrangement:**
- The inequalities are listed on yellow arrows within a blue box.
- The possible solutions are on separate orange pieces on the left and right sides outside the blue box.
- Example: The first inequality \( -4d + 28 \leq 10 \) is matched with \( d < 20 \) as indicated by the pieces being connected.
---
**Note:**
- Solving these inequalities correctly will enhance algebra skills and understanding of inequality solutions.
- This exercise not only involves solving but also matching the correct solutions to improve problem-solving and critical thinking skills.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53fa1db3-32f8-4c94-8cb8-72dc1b15d364%2F920e2dbc-39d5-4bde-a912-83d937d55403%2Ftfj5mai_processed.png&w=3840&q=75)
Step by step
Solved in 4 steps with 4 images
