Pregunta
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  1. Solve the inequality and graph the solutions.

Ask by Gibbs Moreno. in the United States
Jan 24,2025

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All real numbers less than .

Solución

To solve the inequality and graph the solution, follow these steps:

Step 1: Simplify the Inequality

Start by simplifying the right-hand side of the inequality.
Distribute the negative sign inside the parentheses:
Combine like terms ( ):

Step 2: Isolate the Variable Term

Add 3 to both sides to move the constant term to the left side:

Step 3: Solve for

Divide both sides by 16 to isolate :
Simplify the fraction:
This can also be written as:

Step 4: Graph the Solution

To graph the solution on a number line:
  1. Draw a number line with relevant points, including .
  2. Place an open circle at to indicate that is not included in the solution (since the inequality is strictly “<”).
  3. Shade the region to the left of to represent all real numbers less than .
Here is a visual representation:
<----|----------|----------|----------|---->
    0        1/2         1          3/2
          ●---------------->
  • represents the open circle at .
  • The arrow pointing to the left indicates all values less than .

Final Answer

All real numbers less than .

Graph of the Solution

Number Line Graph
(In the graph, an open circle is placed at with shading extending to the left.)

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The Deep Dive

To solve the inequality , let’s first simplify the right side. Distributing the negative sign gives us . Combining like terms results in . Now, add 3 to both sides:
.
Dividing both sides by 16 gives you , or .
To graph the solution, draw a number line and place a circle at with a line extending to the left, indicating all values of that are less than .
Combine those concepts, and voilà! You’ve not only got the solutions but a nice visual too!

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