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Chapter 3: Problem 28

Solve each inequality. $$5(a-2)-6(a+1)>-5$$

Short Answer

Expert verified
a < -11

Step by step solution

01

Distribute constants inside the parentheses

Apply the distributive property to the expression on the left side of the inequality: 5(a-2) - 6(a+1) > -5This results in: 5a - 10 - 6a - 6 > -5
02

Combine like terms

Combine the terms involving 'a' and the constant terms separately: 5a - 6a - 10 - 6 > -5This simplifies to: -a - 16 > -5
03

Isolate the variable term

Add 16 to both sides of the inequality to isolate the term involving 'a': -a - 16 + 16 > -5 + 16 -a > 11
04

Solve for 'a'

To solve for 'a', multiply both sides of the inequality by -1 and reverse the inequality sign (since multiplying by a negative number reverses the inequality): a < -11

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
Understanding the distributive property is essential for solving inequalities. It allows you to multiply a single term across terms inside parentheses. In the inequality \(5(a-2) - 6(a+1) > -5\), you need to distribute the 5 and -6 to both terms inside their respective parentheses.

Applying the distributive property, we get:
  • \( 5(a - 2) \rightarrow 5 \times a + 5 \times (-2) \rightarrow 5a - 10 \)
  • \( -6(a + 1) \rightarrow -6 \times a + (-6) \times 1 \rightarrow -6a - 6 \)
Once distributed, the inequality transforms into: \( 5a - 10 - 6a - 6 > -5 \).

Combining Like Terms
Combining like terms simplifies the expression. In our inequality, we combine similar terms:
  • The terms involving 'a': \( 5a - 6a \) combine to \( -a \)
  • The constant terms: \( -10 - 6 \) combine to \( -16 \)
This simplification transforms our inequality into: \( -a - 16 > -5 \). By combining like terms, you make the inequality easier to handle and prepare it for isolating the variable.

Isolating the Variable
Isolating the variable helps you solve for 'a'. In the inequality \( -a - 16 > -5 \), we isolate 'a' by performing algebraic operations step-by-step:
  • Add 16 to both sides: \( -a - 16 + 16 > -5 + 16 \) simplifies to \( -a > 11 \).
  • Multiply or divide by -1 and remember to reverse the inequality sign: \( -a \times -1 \rightarrow a \) and \( 11 \times -1 \rightarrow -11 \).
    • Once done, the inequality becomes: \( a < -11 \). This shows the range of possible values for 'a' that satisfy the inequality.

    Isolating the variable involves moving all other terms to one side and then solving for the variable. Keep in mind that multiplying or dividing by a negative number reverses the inequality sign!

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Most popular questions from this chapter

Perform the indicated operations and simplify as completely as possible. $$3 a b\left(a^{2}-5 b\right)-3 b\left(a^{3}+4 a b\right)$$

Solve the inequality and sketch the solution set on a number line. $$-3 \leq y-5 \leq 4$$

In Exercises \(47-66,\) solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. Find the width of a rectangle whose length is 6 in. and whose area is 40 sq in.

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