Math Problem Statement

-2(a - 7) - 3a ≥ -31

Solution

Let's solve the given inequality step by step:

Given Inequality:

2(a7)3a31-2(a - 7) - 3a \geq -31

Step 1: Distribute 2-2 into (a7)(a - 7)

2a+(2)(7)=2a+14-2 \cdot a + (-2) \cdot (-7) = -2a + 14 Thus, the inequality becomes: 2a+143a31-2a + 14 - 3a \geq -31

Step 2: Combine like terms

2a3a=5a-2a - 3a = -5a So, the inequality is now: 5a+1431-5a + 14 \geq -31

Step 3: Isolate the term with aa

Subtract 14 from both sides: 5a3114-5a \geq -31 - 14 5a45-5a \geq -45

Step 4: Solve for aa

To isolate aa, divide both sides of the inequality by 5-5. Remember: When dividing or multiplying an inequality by a negative number, the inequality sign flips. a455a \leq \frac{-45}{-5} a9a \leq 9

Final Solution:

a9a \leq 9

Conclusion:

The solution to the inequality is a9a \leq 9.

Would you like more details or have any questions on these steps?

Here are 5 related questions to expand your understanding:

  1. What happens to the inequality symbol when dividing by a negative number?
  2. How do you check if a value satisfies the inequality?
  3. How would the solution change if the inequality was >> instead of \geq?
  4. What are the key differences between solving an equation and an inequality?
  5. Can you graph the solution a9a \leq 9 on a number line?

Tip: When solving inequalities, always remember to reverse the inequality symbol when multiplying or dividing by a negative number!

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities

Formulas

Distributive property
Combining like terms

Theorems

Rules for solving inequalities
Inverse operations

Suitable Grade Level

Grades 7-9