Solve for X: Balancing Inequalities in 8x < 3x + 9 and 5x + 4 < 0

To solve this problem, we will address each inequality separately and examine the conditions:

Step 1: Solving the First Inequality

The first inequality is $8x < 3x + 9$.

• Subtract $3x$ from both sides to isolate terms with $x$:

$8x - 3x < 9$

$5x < 9$

• Divide both sides by 5 to solve for $x$:

$x < \frac{9}{5}$

Thus, the solution to the first inequality is $x < 1.8$.

Step 2: Solving the Second Inequality

The second inequality is $5x + 4 < 0$.

• Subtract 4 from both sides:

$5x < -4$

• Divide both sides by 5:

$x < -\frac{4}{5}$

Thus, $x < -0.8$.

Step 3: Analyzing the Combined Conditions

• We need to find the values of $x$ that satisfy $x < 1.8$ (from Step 1) but do not satisfy $x < -0.8$ (from Step 2).
• This indicates $x$ should be greater than or equal to $-0.8$ and still less than $1.8$.

Therefore, the solution is $-0.8 \leq x < 1.8$.

Thus, the value of $x$ satisfies the desired condition, which corresponds to choice 3 in the options provided.

Therefore, the solution to the problem is $-0.8 \leq x < 1.8$.