Solve: `-7\le2x+1<11`

Question

Which number line is the correct representation of the solution?

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Answer 1

To solve the compound inequality \(-7 \leq 2x + 1 < 11\), we can break it down into two parts:

1. \(-7 \leq 2x + 1\)
2. \(2x + 1 < 11\)

Let's solve each part separately.

Part 1: \(-7 \leq 2x + 1\)

Subtract 1 from both sides:

\[ -7 - 1 \leq 2x \]

\[ -8 \leq 2x \]

Now, divide both sides by 2 (note that we do not flip the inequality here since we are dividing by a positive number):

\[ -4 \leq x \]

This can also be written as:

\[ x \geq -4 \]

Part 2: \(2x + 1 < 11\)

Subtract 1 from both sides:

\[ 2x < 11 - 1 \]

\[ 2x < 10 \]

Now, divide both sides by 2:

\[ x < 5 \]

Combining the Results

Now we can combine our results from both parts:

\[ -4 \leq x < 5 \]

Number Line Representation

To represent the solution \(-4 \leq x < 5\) on a number line:

• Place a closed circle on -4 (indicating that it is included).
• Place an open circle on 5 (indicating that it is not included).
• Shade the region between -4 and 5.

This gives us the correct representation of the solution on the number line.

Answer 2

12x - 3 less than equal too 15x or -0.2x + 10 < 5

Which choice best describes the solution set?

1\le x<-25

-1\le x<25

x\ge1 or <-25

x\le-1 or x>25

Answer 3

To solve the inequalities you've provided, let's analyze them one at a time.

First Inequality: \( 12x - 3 \leq 15x \)

1. Start by isolating the variable \(x\): \[ 12x - 3 \leq 15x \] Subtract \(12x\) from both sides: \[ -3 \leq 3x \] Now, divide both sides by 3: \[ -1 \leq x \] This can be rewritten as: \[ x \geq -1 \]

Second Inequality: \( -0.2x + 10 < 5 \)

1. Start by isolating the variable \(x\): \[ -0.2x + 10 < 5 \] Subtract 10 from both sides: \[ -0.2x < 5 - 10 \] Which simplifies to: \[ -0.2x < -5 \] Now, divide both sides by -0.2. Remember to flip the inequality sign when dividing by a negative number: \[ x > \frac{-5}{-0.2} = 25 \] Thus, we have: \[ x > 25 \]

Combining the Results

Now we need to combine the results of the two inequalities:

1. The first inequality gives us \(x \geq -1\).
2. The second inequality gives us \(x > 25\).

The solution set includes both conditions:

\[ x \geq -1 \quad \text{or} \quad x > 25 \]

Evaluating the Choices

Now let's evaluate the choices based on what we've derived:

1. \(1 \le x < -25\) - Incorrect
2. \(-1 \le x < 25\) - Incorrect
3. \(x \ge 1\) or \(x < -25\) - Incorrect
4. \(x \le -1\) or \(x > 25\) - Incorrect

From our solution, the correct format would be:

\[ x \ge -1 \quad \text{or} \quad x > 25 \]

Since none of the provided choices perfectly match our derived answer, there might be an oversight in the options listed. Based on the solutions:

• The correct answer from the given conditions could be summarized as \(x \ge -1\) or \(x > 25\), which isn't listed. Assuming the options are correctly specified, select the option that best aligns with individual inequalities.

If you have an option that states \(x \ge -1\) or \(x > 25\) that should be chosen.