Which graph below bests represent the solution set for `-2x+5+x\le3x+13`?.....

Answer 1

To solve the inequality \(-2x + 5 + x \leq 3x + 13\), we first combine like terms:

1. Combine terms on the left side: \[ -2x + x + 5 \leq 3x + 13 \] This simplifies to: \[ -x + 5 \leq 3x + 13 \]

2. Now, we will isolate \(x\). First, we can add \(x\) to both sides: \[ 5 \leq 4x + 13 \]

3. Next, subtract \(13\) from both sides: \[ 5 - 13 \leq 4x \] This gives us: \[ -8 \leq 4x \]

4. Now divide both sides by \(4\): \[ -\frac{8}{4} \leq x \] Which simplifies to: \[ -2 \leq x \] or equivalently: \[ x \geq -2 \]

The solution set for the inequality is \(x \geq -2\), which can be represented graphically as a number line where all numbers greater than or equal to \(-2\) are included.

In a graphing representation, this would look like a filled (or solid) point at \(-2\) with a line extending to the right towards positive infinity. Look for a graph that represents that.

Answer 2

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

Which number line is the correct representation of the solution?

Answer 3

To solve the compound inequality \(-7 \leq 2x + 1 < 11\), we will break it down into two parts and solve each part step-by-step.

Step 1: Solve the left side of the compound inequality

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

Subtract \(1\) from both sides:

\[ -7 - 1 \leq 2x \] \[ -8 \leq 2x \]

Now, divide both sides by \(2\):

\[ -4 \leq x \] or equivalently, \[ x \geq -4 \]

Step 2: Solve the right side of the compound inequality

\[ 2x + 1 < 11 \]

Subtract \(1\) from both sides:

\[ 2x < 11 - 1 \] \[ 2x < 10 \]

Now, divide both sides by \(2\):

\[ x < 5 \]

Step 3: Combine the two parts

Putting it all together, we have:

\[ -4 \leq x < 5 \]

Final Result

The solution for the given compound inequality is:

\[ -4 \leq x < 5 \]

Representation on a Number Line

On a number line, this solution can be represented as:

• A closed circle (or solid point) on \(-4\) (to indicate that \(-4\) is included in the solution).
• An open circle (or hollow point) on \(5\) (to indicate that \(5\) is not included in the solution).
• A line extending from \(-4\) to \(5\).

Look for a number line graph that has these characteristics:

• Closed (filled) dot at \(-4\)
• Open (unfilled) dot at \(5\)
• A line connecting them, indicating that all numbers in this interval are included in the solution.