To solve the equation [tex]\(-4 - \frac{2}{3} x = -6\)[/tex], let's follow a step-by-step approach:
1. Adding 4 to both sides:
[tex]\[
-4 - \frac{2}{3} x + 4 = -6 + 4
\][/tex]
This simplifies to:
[tex]\[
-\frac{2}{3} x = -2
\][/tex]
So, the equation after adding 4 to both sides is indeed:
[tex]\[
-\frac{2}{3} x = -2
\][/tex]
2. Checking the proposed steps:
- The statement "After adding 4 to both sides, the equation is [tex]\(-\frac{2}{3} x = -2\)[/tex]" is correct.
- The statement "After adding 4 to both sides, the equation is [tex]\(-\frac{2}{3} x = -10\)[/tex]" is incorrect, as shown by the above simplification.
3. Solving for [tex]\(x\)[/tex] using one more step:
- To isolate [tex]\(x\)[/tex], we can either multiply both sides by the reciprocal of [tex]\(-\frac{2}{3}\)[/tex], which is [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[
(-\frac{2}{3} x) \cdot (-\frac{3}{2}) = (-2) \cdot (-\frac{3}{2})
\][/tex]
Simplifying this, we get:
[tex]\[
x = 3
\][/tex]
- Alternatively, we can divide both sides of the equation by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[
\frac{-\frac{2}{3} x}{-\frac{2}{3}} = \frac{-2}{-\frac{2}{3}}
\][/tex]
This simplifies to:
[tex]\[
x = 3
\][/tex]
Based on these options:
- 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 correct.
- 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 also correct.
- The statement "The equation can be solved for [tex]\(x\)[/tex] using exactly one more step by multiplying both sides by [tex]\(-\frac{2}{3}\)[/tex]" is incorrect.
So, the three true statements are:
1. After adding 4 to both sides, the equation is [tex]\(-\frac{2}{3} x = -2\)[/tex].
2. The equation can be solved for [tex]\(x\)[/tex] using exactly one more step by multiplying both sides by [tex]\(-\frac{3}{2}\)[/tex].
3. The equation can be solved for [tex]\(x\)[/tex] using exactly one more step by dividing both sides by [tex]\(-\frac{2}{3}\)[/tex].
