Certainly! Let's solve the inequality [tex]\(4x + 5 \geq 13\)[/tex] step-by-step and then describe how to graph it.
### Step 1: Isolate the variable term
Start with the original inequality:
[tex]\[ 4x + 5 \geq 13 \][/tex]
To isolate the term with the variable [tex]\(x\)[/tex], subtract 5 from both sides of the inequality:
[tex]\[ 4x + 5 - 5 \geq 13 - 5 \][/tex]
[tex]\[ 4x \geq 8 \][/tex]
### Step 2: Solve for the variable
Next, divide both sides of the inequality by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{4x}{4} \geq \frac{8}{4} \][/tex]
[tex]\[ x \geq 2 \][/tex]
Thus, the solution to the inequality is:
[tex]\[ x \geq 2 \][/tex]
### Step 3: Graphing the inequality
To graph the inequality [tex]\(4x + 5 \geq 13\)[/tex]:
1. Draw the boundary line: First, graph the equation [tex]\(4x + 5 = 13\)[/tex].
- This simplifies to [tex]\(4x = 8\)[/tex], so the boundary line is [tex]\(x = 2\)[/tex].
- On a number line, draw a vertical line at [tex]\(x = 2\)[/tex].
2. Determine the region to shade:
- Since the inequality is [tex]\(x \geq 2\)[/tex], you will shade the region to the right of [tex]\(x = 2\)[/tex].
3. Check your shading:
- To confirm, pick a test point from each side of the boundary line [tex]\(x = 2\)[/tex]. For instance, test [tex]\(x = 0\)[/tex] and [tex]\(x = 3\)[/tex].
- If [tex]\(x = 0\)[/tex]: [tex]\(4(0) + 5 = 5\)[/tex], which is not [tex]\(\geq 13\)[/tex]. So, [tex]\(x = 0\)[/tex] is not a solution.
- If [tex]\(x = 3\)[/tex]: [tex]\(4(3) + 5 = 17\)[/tex], which is [tex]\(\geq 13\)[/tex]. So, [tex]\(x = 3\)[/tex] is a solution.
### Summary of the inequality solution and its graph:
The solution to the inequality [tex]\(4x + 5 \geq 13\)[/tex] is [tex]\(x \geq 2\)[/tex].
- On a number line:
- Draw a circle at [tex]\(x = 2\)[/tex] and fill it in (to show that [tex]\(2\)[/tex] is included in the solution).
- Shade the number line to the right of [tex]\(x = 2\)[/tex] to indicate all values [tex]\(x\)[/tex] that are greater than or equal to [tex]\(2\)[/tex].
Now you have both solved and graphed the inequality effectively!
