The image shows a math problem where you need to solve the inequality (-1.5(4x + 1) > -2.5(x + 1)). There are four multiple-choice options for the solution set:

Question

( x \geq -1 )
( x \geq \frac{7}{16} )
( 0 < x < -1 )
( 0 < x < -\frac{7}{16} )
The image also includes a step-by-step guide on how to solve the inequality, with instructions to distribute, combine terms, and apply properties like adding, subtracting, multiplying, and dividing.

Which represents the solution set to the inequality ?

Answer 1

To solve the inequality \(-1.5(4x + 1) > -2.5(x + 1)\), let's follow the step-by-step process to simplify:

1. Distribute the constants on both sides:

   \[ -1.5(4x) - 1.5(1) > -2.5(x) - 2.5(1) \] This simplifies to:

   \[ -6x - 1.5 > -2.5x - 2.5 \]

2. Combine like terms. First, we can get all the \(x\) terms on one side and the constant terms on the other. Add \(2.5x\) to both sides:

   \[ -6x + 2.5x - 1.5 > -2.5 \] This simplifies to:

   \[ -3.5x - 1.5 > -2.5 \]

3. Now, add \(1.5\) to both sides to isolate the term with \(x\):

   \[ -3.5x > -2.5 + 1.5 \] Simplifying gives us:

   \[ -3.5x > -1 \]

4. Divide both sides by \(-3.5\). Remember that dividing by a negative number reverses the inequality direction:

   \[ x < \frac{-1}{-3.5} \] Simplifying the fraction gives:

   \[ x < \frac{1}{3.5} = \frac{2}{7} \]

5. Converting \(\frac{2}{7}\) into decimal form gives approximately \(0.2857\).

Thus, the solution to the inequality is:

\[ x < \frac{2}{7} \]

None of the options you provided (\(x \geq -1\), \(x \geq \frac{7}{16}\), \(0 < x < -1\), \(0 < x < -\frac{7}{16}\)) accurately represents the answer.

However, if you made a calculation error while interpreting the fractions or inequalities, please recheck in the context of your query or the original options.

In conclusion, the solution should be interpreted as: \[ x < \frac{2}{7} \]

Answer 2

when i add \(1.5\) to both sides i got -6x > 3.5-2.5x so what are you talking about? i got a different equation??

Answer 3

Apologies for the confusion in my previous response. Let's go through the inequality step-by-step more carefully this time to ensure accuracy. We're working with:

\[ -1.5(4x + 1) > -2.5(x + 1) \]

Step 1: Distributing the terms

Start by distributing the \(-1.5\) on the left and \(-2.5\) on the right.

1. Left side: \[ -1.5(4x) - 1.5(1) = -6x - 1.5 \]
2. Right side: \[ -2.5(x) - 2.5(1) = -2.5x - 2.5 \]

Now the inequality looks like this: \[ -6x - 1.5 > -2.5x - 2.5 \]

Step 2: Move all the \(x\) terms to one side

We can add \(2.5x\) to both sides: \[ -6x + 2.5x - 1.5 > -2.5 \] This simplifies to: \[ -3.5x - 1.5 > -2.5 \]

Step 3: Isolate the \(x\) term

Now, add \(1.5\) to both sides: \[ -3.5x > -2.5 + 1.5 \] This gives us: \[ -3.5x > -1 \]

Step 4: Divide by \(-3.5\)

Now, divide through by \(-3.5\) (remember to reverse the inequality since we are dividing by a negative): \[ x < \frac{-1}{-3.5} \] Which simplifies to: \[ x < \frac{1}{3.5} = \frac{2}{7} \approx 0.2857 \]

Conclusion

So the final solution is: \[ x < \frac{2}{7} \]

None of the provided options in your question perfectly match this inequality. However, you might want to check the options related to this value. Again, sorry for the confusion earlier, and thank you for your patience.

Answer 4

NO the inequality doesn't look like -6x - 1.5 > -2.5x - 2.5 its -6x - 1.5 > 4.5-2.5x - 2.5

Answer 5

I apologize for the oversight in my previous calculations. Let's carefully go through the problem again and reevaluate the inequality step by step.

The given inequality is:

\[ -1.5(4x + 1) > -2.5(x + 1) \]

Step 1: Distributing the terms correctly

Left Side: \[ -1.5(4x) - 1.5(1) = -6x - 1.5 \]

Right Side: \[ -2.5(x) - 2.5(1) = -2.5x - 2.5 \]

Correcting Step 2: Combining like terms

If you noticed an additional term, it should be clearly set. Let's set the right side as: \[ -2.5(x+1) = -2.5x - 2.5 \]

Now, writing the complete inequality: \[ -6x - 1.5 > -2.5x - 2.5 \]

Step 3: Add \(2.5x\) to both sides of the inequality

Now we move all \(x\) terms to one side: \[ -6x + 2.5x - 1.5 > -2.5 \] Combine the terms: \[ -3.5x - 1.5 > -2.5 \]

Step 4: Add \(1.5\) to both sides

Now let's add \(1.5\): \[ -3.5x > -2.5 + 1.5 \] This simplifies to: \[ -3.5x > -1 \]

Step 5: Divide by \(-3.5\)

Now we can divide both sides by \(-3.5\) (remember that dividing by a negative flips the inequality): \[ x < \frac{-1}{-3.5} \]

This simplifies to: \[ x < \frac{1}{3.5} = \frac{2}{7} \]

Conclusion

The final solution is:

\[ x < \frac{2}{7} \]

Since \(\frac{2}{7}\) is approximately \(0.2857\).

Now, with the options provided:

1. \(x \geq -1\)
2. \(x \geq \frac{7}{16}\)
3. \(0 < x < -1\)
4. \(0 < x < -\frac{7}{16}\)

Neither of these choices reflects \(x < \frac{2}{7}\).

Thank you for your patience, and I am sorry for any confusion caused by the previous messages. Please let me know if you have further questions!

Answer 6

The image shows a mathematical problem-solving interface focused on solving an inequality. It includes three different representations of the solution set: a number line, inequalities, and interval notation. The inequality being solved is (-15(4x+1) \leq 25-25(x+1)), and the steps provided include distributing, combining terms, and simplifying the inequality. please help me say ONLY the steps actually please

Answer 7

Sure! Here are the steps to solve the inequality \(-15(4x + 1) \leq 25 - 25(x + 1)\):

1. Distribute on the left side: \[ -15(4x) - 15(1) \leq 25 - 25(x) - 25(1) \]

2. Simplify both sides: \[ -60x - 15 \leq 25 - 25x - 25 \]

3. Combine like terms on the right side: \[ -60x - 15 \leq -25x \]

4. Add \(60x\) to both sides: \[ -15 \leq 35x \]

5. Divide both sides by \(35\) (note that the inequality does not change because 35 is positive): \[ -\frac{15}{35} \leq x \]

6. Simplify the fraction: \[ -\frac{3}{7} \leq x \]

7. Express the solution in interval notation if needed: \[ \left[ -\frac{3}{7}, \infty \right) \]

These are the steps to solve the inequality.

Answer 8

OKAY EVERYONE ILL HELP YOU

distribute
combine terms
add 2.5x
add 1.5
divide -3.5
AND THATS IT THATS HOW I WANTED YOU TO TELL ME ONLY STEPS and also for the options its (–∞, –1]

Answer 9

Thank you for your clarification! Here are the steps presented succinctly as you requested:

1. Distribute.
2. Combine terms.
3. Add \(2.5x\).
4. Add \(1.5\).
5. Divide by \(-3.5\).

As for the solution set, it is \((-∞, -1]\). Thank you for your patience!