Let's solve the equation step by step and evaluate which statements are true.
The given equation is:
[tex]\[ -4 - \frac{2}{3}x = -6 \][/tex]
Step 1: Add 4 to both sides
[tex]\[ -4 - \frac{2}{3}x + 4 = -6 + 4 \][/tex]
Simplify both sides:
[tex]\[ -\frac{2}{3}x = -2 \][/tex]
This means that the statement "After adding 4 to both sides, the equation is [tex]\(-\frac{2}{3} x = -2\)[/tex]" is true.
Step 2a: Multiply both sides by [tex]\(-\frac{3}{2}\)[/tex]
We want to isolate [tex]\(x\)[/tex], so we can multiply both sides by the reciprocal of [tex]\(-\frac{2}{3}\)[/tex], which is [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ \left(-\frac{3}{2}\right) \left(-\frac{2}{3}x \right) = \left(-\frac{3}{2}\right) (-2) \][/tex]
Simplify both sides:
[tex]\[ x = 3 \][/tex]
This means that 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.
Step 2b: Divide both sides by [tex]\(-\frac{2}{3}\)[/tex]
Alternatively, we can solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[ \frac{-\frac{2}{3}x}{-\frac{2}{3}} = \frac{-2}{-\frac{2}{3}} \][/tex]
Simplify both sides:
[tex]\[ x = 3 \][/tex]
This means that 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.
Now let us evaluate the rest of the 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)
Therefore, the three true statements are:
1. After adding 4 to both sides, the equation is [tex]\(-\frac{2}{3} x = -2\)[/tex].
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].
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].
