To solve the inequality [tex]\(4.2x + 5.6 < 7.2 - 8.3x\)[/tex], let's follow a step-by-step approach:
### Step 1: Isolate the constant term on the left side of the inequality
First, we'll move all terms involving [tex]\(x\)[/tex] to one side and all constant terms to the other side.
To do this, add [tex]\(8.3x\)[/tex] to both sides of the inequality:
[tex]\[4.2x + 8.3x + 5.6 < 7.2 - 8.3x + 8.3x\][/tex]
This simplifies to:
[tex]\[12.5x + 5.6 < 7.2\][/tex]
Now, subtract [tex]\(5.6\)[/tex] from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[12.5x < 7.2 - 5.6\][/tex]
### Step 2: Simplify the constants
Simplify the constants on the right side:
[tex]\[12.5x < 1.6\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], divide both sides of the inequality by [tex]\(12.5\)[/tex]:
[tex]\[x < \frac{1.6}{12.5}\][/tex]
Simplifying the fraction gives:
[tex]\[x < 0.128\][/tex]
### Step 4: Write the solution in interval notation
The solution to the inequality is:
[tex]\[(-\infty, 0.128)\][/tex]
This means [tex]\(x\)[/tex] can be any value less than [tex]\(0.128\)[/tex].
