Solve and graph: -16 ≤ 3x - 4<2

Answer 1

To solve the compound inequality \(-16 \leq 3x - 4 < 2\), we can break it down into two parts and solve each part step-by-step.

Step 1: Solve the left part of the inequality

Inequality: \(-16 \leq 3x - 4\)

1. Add 4 to both sides: \[ -16 + 4 \leq 3x \] \[ -12 \leq 3x \]

2. Divide by 3: \[ -4 \leq x \] This can be rewritten as: \[ x \geq -4 \]

Step 2: Solve the right part of the inequality

Inequality: \(3x - 4 < 2\)

1. Add 4 to both sides: \[ 3x - 4 + 4 < 2 + 4 \] \[ 3x < 6 \]

2. Divide by 3: \[ x < 2 \]

Step 3: Combine the results

From both parts, we have: \[ -4 \leq x < 2 \]

Step 4: Write the solution in interval notation

The solution can be expressed in interval notation as: \[ [-4, 2) \]

Step 5: Graph the solution

To graph the solution, we represent the interval on a number line:

• Place a solid dot at \(-4\) (since \(-4\) is included in the solution).
• Place an open dot at \(2\) (since \(2\) is not included in the solution).
• Shade the area between \(-4\) and \(2\).

Here is a rough representation of the graph:

  <----|----|----|----|----|----|----|----|----|---->
       -5  -4   -3   -2   -1    0    1    2    3

         ●==================
            -4              2

Remember, the solid dot at \(-4\) indicates that \(-4\) is included in the solution, while the open dot at \(2\) indicates that \(2\) is not included in the solution.