Let's solve the equation step-by-step to determine which statements could be true.
We start with the equation:
[tex]\[ -4 - \frac{2}{3}x = -6 \][/tex]
Step 1: Add 4 to both sides.
Adding 4 to both sides results in:
[tex]\[ -4 + 4 - \frac{2}{3}x = -6 + 4 \][/tex]
Simplifying, we get:
[tex]\[ -\frac{2}{3}x = -2 \][/tex]
So, the statement "After adding 4 to both sides, the equation is [tex]\( -\frac{2}{3}x = -2 \)[/tex]" is true.
Step 2: Solve for [tex]\( x \)[/tex].
To isolate [tex]\( x \)[/tex], we can either multiply or divide by the coefficient of [tex]\( x \)[/tex]. Let's explore these options.
Option 1: Multiply both sides by the reciprocal of the coefficient.
The coefficient of [tex]\( x \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex]. Its reciprocal is [tex]\( -\frac{3}{2} \)[/tex]. Thus, we multiply both sides by [tex]\( -\frac{3}{2} \)[/tex]:
[tex]\[ x = (-2) \times \left(-\frac{3}{2}\right) \][/tex]
[tex]\[ x = 3 \][/tex]
So, the statement "The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by multiplying both sides by [tex]\( -\frac{3}{2} \)[/tex]" is true.
Option 2: Divide both sides by the coefficient of [tex]\( x \)[/tex].
Alternatively, we can divide both sides by [tex]\( -\frac{2}{3} \)[/tex]:
[tex]\[ x = \frac{-2}{-\frac{2}{3}} \][/tex]
[tex]\[ x = -2 \div -\frac{2}{3} \][/tex]
[tex]\[ x = -2 \times -\frac{3}{2} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the statement "The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by dividing both sides by [tex]\( -\frac{2}{3} \)[/tex]" is true.
Checking the given statements:
1. After adding 4 to both sides, the equation is [tex]\( -\frac{2}{3}x = -2 \)[/tex]. True
2. After adding 4 to both sides, the equation is [tex]\( -\frac{2}{3}x = -10 \)[/tex]. False
3. The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by multiplying both sides by [tex]\( -\frac{3}{2} \)[/tex]. True
4. The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by dividing both sides by [tex]\( -\frac{2}{3} \)[/tex]. True
5. The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by multiplying both sides by [tex]\( -\frac{2}{3} \)[/tex]. False, because multiplying by [tex]\( -\frac{2}{3} \)[/tex] would complicate the equation further.
Conclusion:
The three true statements are:
- After adding 4 to both sides, the equation is [tex]\( -\frac{2}{3}x = -2 \)[/tex].
- The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by multiplying both sides by [tex]\( -\frac{3}{2} \)[/tex].
- The equation can be solved for [tex]\( x \)[/tex] using exactly one more step by dividing both sides by [tex]\( -\frac{2}{3} \)[/tex].
