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What is the solution set of x+2≥6 and 3x≥6 ?(1 point)

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9 answers

To find the solution set for the inequalities \( x + 2 \geq 6 \) and \( 3x \geq 6 \), we will solve each inequality separately.

  1. Solve \( x + 2 \geq 6 \): \[ x + 2 \geq 6 \] Subtract 2 from both sides: \[ x \geq 4 \]

  2. Solve \( 3x \geq 6 \): \[ 3x \geq 6 \] Divide both sides by 3: \[ x \geq 2 \]

Now we have two inequalities:

  • From the first inequality, \( x \geq 4 \)
  • From the second inequality, \( x \geq 2 \)

The solution set that satisfies both inequalities is the intersection of \( x \geq 4 \) and \( x \geq 2 \). Since \( x \geq 4 \) is the more restrictive condition, the final solution set is:

\[ { x \mid x \geq 4 } \]

In interval notation, this can be expressed as: \[ [4, \infty) \]

What is the solution set of 2x−3=5 and 5x+3>−12 ?(1 point) put it in a way that is less complicated

Let's solve the equations and inequalities step by step to find the solution set.

  1. Solve the equation \( 2x - 3 = 5 \): \[ 2x - 3 = 5 \] Add 3 to both sides: \[ 2x = 8 \] Divide by 2: \[ x = 4 \]

  2. Solve the inequality \( 5x + 3 > -12 \): \[ 5x + 3 > -12 \] Subtract 3 from both sides: \[ 5x > -15 \] Divide by 5: \[ x > -3 \]

Now we have:

  • From the equation: \( x = 4 \)
  • From the inequality: \( x > -3 \)

Conclusion: The solution set that meets both conditions is just the single point where the equation is satisfied, which is \( x = 4 \).

Thus, the final answer can be expressed as: \[ { x \mid x = 4 } \]

Or simply: \[ { 4 } \]

Use the image to answer the question.

A number line ranges from negative 5 to 5 in increments of 1. An open point is plotted at 2, with a solid line extending right, beyond 5. An open point is plotted at 0, with a dotted line extending left, beyond negative 5.

What is the solution set to the inequalities on the graph?

(1 point)
x<
or x>

Based on your description of the number line:

  1. There is an open point at 2, with a solid line extending right beyond 5. This indicates that the inequality is: \[ x > 2 \]

  2. There is an open point at 0, with a dotted line extending left beyond -5. This indicates that the inequality is: \[ x < 0 \]

Therefore, the solution set for the inequalities represented on the graph is: \[ x < 0 \quad \text{or} \quad x > 2 \]

In interval notation, this can be expressed as: \[ (-\infty, 0) \cup (2, \infty) \]

So, the final response to your question is: \[ x < 0 \quad \text{or} \quad x > 2 \]

Use the image to answer the question.

A number line extends from negative 5 to 5 in increments of 1.

To indicate the solution for x≥−1 and x<4 , where would you place circles on the graph? Enter the lesser value first.

(1 point)
and

Based on the inequalities \( x \geq -1 \) and \( x < 4 \):

  1. For \( x \geq -1 \): You would place a solid circle (or closed circle) on -1 to indicate that -1 is included in the solution set.

  2. For \( x < 4 \): You would place an open circle on 4 to indicate that 4 is not included in the solution set.

So, you would place the circles as follows:

  • Solid circle at \(-1\)
  • Open circle at \(4\)

In terms of just placing the numbers:

  • \(-1\) (with a solid circle)
  • \(4\) (with an open circle)

Thus, entering the lesser value first, you would write: \[ -1 \text{ and } 4 \]

Use the image to answer the question.

A number line ranges from negative 3 to 8 in increments of 1. Two closed points are plotted at negative 2 and 7, with a solid line extending between them.

What solution set does the graph represent?

(1 point)
x≤
and x≥

Based on your description of the number line:

  1. Closed points at -2 and 7 indicate that both -2 and 7 are included in the solution set.

  2. A solid line extending between -2 and 7 indicates all the values between -2 and 7 are included.

This corresponds to the following inequalities:

  • Since there is a closed point at -2, we have \( x \geq -2 \).
  • Since there is a closed point at 7, we have \( x \leq 7 \).

Thus, the solution set represented by the graph is: \[ x \geq -2 \quad \text{and} \quad x \leq 7 \]

In summary, the answer is:

  • \( x \geq -2 \)
  • \( x \leq 7 \)

You can express the interval notation for this solution set as: \[ [-2, 7] \]