Question

solve the inequality. explain how you found your answer. 6(x - 2) - 3 ≥ -3(-2 + 9) + 7 (type an integer or a fraction.) what is the best next step to solve the given inequality? a. apply the addition or subtraction property of inequality b. combine like terms where possible c. apply the distributive property. d. apply the multiplication or division property of inequality. apply the distributive property, and simplify 6(x - 2) - 3 ≥ -14 □ ≥ -14 (simplify your answer.) enter your answer in the answer box and then click check answer

Explanation:

Step1: Apply Distributive Property

First, apply the distributive property to \(6(x - 2)\). The distributive property states that \(a(b - c)=ab - ac\), so \(6(x - 2)=6x-12\).

Step2: Simplify Left Side

Now, substitute back into the inequality: \(6x - 12-3\geq - 14\). Combine the constant terms on the left side: \(6x-(12 + 3)=6x-15\). Wait, but the problem has a box after \(6(x - 2)-3\) simplified. Wait, let's re - evaluate. Wait, the original left side is \(6(x - 2)-3\). After applying distributive property: \(6x-12 - 3\). Combine like terms (the constants - 12 and - 3): \(6x-(12 + 3)=6x-15\)? Wait, no, the right side was calculated as \(-3(-2 + 9)+7=-3(7)+7=-21 + 7=-14\). Then the left side after distributive property: \(6(x - 2)-3=6x-12-3\). Now, combine the - 12 and - 3: \(6x-15\)? But the problem shows \(6(x - 2)-3\) simplifies to a boxed expression \(\geq - 14\). Wait, maybe I made a mistake. Wait, \(6(x - 2)-3=6x-12 - 3=6x-15\)? No, wait, let's do it again. Wait, \(6(x - 2)=6x-12\), then \(6x-12-3=6x-(12 + 3)=6x-15\). But maybe the problem is in the middle of the process. Wait, the user's problem has "Apply the Distributive Property, and simplify \(6(x - 2)-3\geq - 14\) \(\square\geq - 14\)". So we need to simplify \(6(x - 2)-3\). Let's expand \(6(x - 2)\) to get \(6x-12\), then subtract 3: \(6x-12-3=6x-15\). Wait, but maybe the question is to simplify \(6(x - 2)-3\) after applying distributive property. So \(6(x - 2)-3 = 6x-12-3=6x-15\). But let's check the right side calculation: \(-3(-2 + 9)+7=-3(7)+7=-21 + 7=-14\). So the inequality becomes \(6x-15\geq - 14\)? Wait, no, maybe I messed up the right - hand side calculation. Wait, \(-3(-2 + 9)=-3(7)=-21\), then \(-21 + 7=-14\). That's correct. Then left side: \(6(x - 2)-3=6x-12-3=6x-15\). So the simplified left side is \(6x - 15\). But the problem's box is after \(6(x - 2)-3\) is simplified. Wait, maybe the user made a typo, but according to the distributive property and combining like terms, \(6(x - 2)-3=6x-12-3=6x-15\). But let's see, maybe the next step is to solve for \(x\), but the question here is to simplify \(6(x - 2)-3\). So \(6(x - 2)-3=6x-12-3=6x-15\). But let's check again. Wait, \(6(x - 2)=6x-12\), then \(6x-12-3=6x-(12 + 3)=6x-15\). So the box should be filled with \(6x - 15\). But maybe the problem is in the process of solving, and the next step after applying distributive property is to simplify the left - hand side by combining like terms. Wait, the first part of the question is "What is the best next step to solve the given inequality?". The inequality is \(6(x - 2)-3\geq - 3(-2 + 9)+7\). First, we can simplify the right - hand side: \(-3(-2 + 9)+7=-3(7)+7=-21 + 7=-14\). Then the left - hand side has a parenthesis, so the best next step is to apply the distributive property to \(6(x - 2)\), which is option C. Then, after applying the distributive property, we have \(6x-12-3\geq - 14\), and then combine like terms ( - 12 and - 3) to get \(6x-15\geq - 14\). So the simplified left - hand side is \(6x - 15\).

Answer:

For the "What is the best next step..." question, the answer is C. Apply the Distributive Property.

For the "Apply the Distributive Property, and simplify" part, the answer is \(6x - 15\).