Image to Text

Question

Graph the solution of this inequality:
[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3
]

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Drag a point to the number line.

Summary

The image presents an inequality problem: (\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3). Below the inequality, there is a number line ranging from -7 to 7. There are also draggable points with arrows indicating possible directions, suggesting that the user can place these points on the number line to graph the solution of the inequality.

Answer 1

To solve the inequality

\[ \frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3, \]

we first simplify the left side:

1. Distribute \(\frac{3}{7}\): \[ \frac{3}{7} \times 35x - \frac{3}{7} \times 14 = 15x - 6. \] Thus, the left side simplifies to \(15x - 6\).

2. Rewrite the inequality: \[ 15x - 6 \leq \frac{21x}{2} + 3. \]

3. Next, we need to eliminate the fraction on the right side. We can multiply the entire inequality by 2 (which does not change the inequality sign as we are multiplying by a positive number): \[ 2(15x - 6) \leq 21x + 6. \]

   This simplifies to: \[ 30x - 12 \leq 21x + 6. \]

4. Now, we isolate \(x\). Subtract \(21x\) from both sides: \[ 30x - 21x - 12 \leq 6, \] which simplifies to: \[ 9x - 12 \leq 6. \]

5. Then, add 12 to both sides: \[ 9x \leq 18. \]

6. Finally, divide both sides by 9: \[ x \leq 2. \]

So, the solution to the inequality is \(x \leq 2\).

Graphing the solution

To graph the solution on the number line from -7 to 7:

• Draw a closed circle at \(2\) (indicating that \(2\) is included in the solution).
• Shade the region to the left of \(2\) to indicate all values less than or equal to \(2\).

Final Step

You can drag a point to the number line at \(2\) and shade to the left to graph the solution \(x \leq 2\).

Answer 2

CLOSED POINT GOING THIS WAY <---- ON 2 NOT -2 THAnK YOU AI now What is the solution set for the inequality?

(–∞, 2]
(–∞, ]
[2, ∞)
[, ∞)

Answer 3

The solution set for the inequality \(x \leq 2\) is expressed in interval notation as:

\[ (-\infty, 2] \]

This means that \(x\) can take any value from negative infinity up to and including \(2\).

So the correct choice from your options is:

(–∞, 2]

Answer 4

HOORAY!!

Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?

x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)

Answer 5

To solve the inequality

\[ 5.1(3 + 2.2x) > -14.25 - 6(1.7x + 4), \]

let's break it down step by step.

1. Distribute on both sides:

   Left side: \[ 5.1 \times 3 + 5.1 \times 2.2x = 15.3 + 11.22x. \]

   Right side: \[ -14.25 - 6(1.7x + 4) = -14.25 - 10.2x - 24 = -14.25 - 10.2x - 24 = -38.25 - 10.2x. \]

   So, rewriting the inequality gives us: \[ 15.3 + 11.22x > -38.25 - 10.2x. \]

2. Combine like terms:

   First, add \(10.2x\) to both sides: \[ 15.3 + 11.22x + 10.2x > -38.25. \] This simplifies to: \[ 15.3 + 21.42x > -38.25. \]

3. Now, isolate the term with \(x\):

   Subtract \(15.3\) from both sides: \[ 21.42x > -38.25 - 15.3. \] This evaluates to: \[ 21.42x > -53.55. \]

4. Divide both sides by \(21.42\): \[ x > \frac{-53.55}{21.42}. \] Calculating the right-hand side gives: \[ x > -2.5. \]

Conclusion:

The solution of the inequality is:

\[ x > -2.5. \]

In interval notation, the solution set is:

\[ (-2.5, \infty). \]

So, the correct answer choice from your options is:

(–2.5, ∞).