Problem 21 For Problems $1-50$, solve eac... [FREE SOLUTION]

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Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 3: Problem 21

For Problems $1-50$, solve each inequality. (Objectives 1 and 2) $$ -2(x+6)>-17 $$

Short Answer

Expert verified

The solution is $x < \frac{5}{2}$.

Step by step solution

Distribute the Negative Sign

Start by distributing the -2 in the inequality $-2(x+6) > -17$. This gives us $-2x - 12 > -17$. Be careful to apply the negative sign to both terms inside the parenthesis.

Isolate Variable Term

To isolate the $-2x$ term, add 12 to both sides of the inequality: $-2x - 12 + 12 > -17 + 12$. This simplifies to $-2x > -5$.

Solve for x

Now, divide both sides of the inequality by -2. Remember, dividing by a negative 鈥渞everselqeqes鈥� the inequality symbol: $x < \frac{5}{2}$ or $x < 2.5$.

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Key Concepts

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

Distributive Property

The distributive property is a fundamental principle in algebra used to multiply a single term across terms in a bracket. In the inequality $-2(x+6) > -17$, we see the distributive property in action. Here, the $-2$ is multiplied by each term inside the parenthesis:

Multiply $-2$ by $x$ to get $-2x$.
Multiply $-2$ by $6$ to get $-12$.

So, we rewrite the inequality as \[-2x - 12 > -17\].
This step ensures that each term inside the parenthesis is individually affected by the factor outside.
It helps in simplifying expressions by getting rid of brackets and making subsequent calculations straightforward.

Isolation of Variables

Isolation of the variable is a crucial step in solving both equations and inequalities. We aim to keep only the variable on one side of the inequality. In our example, with $-2x - 12 > -17$, we need to isolate $-2x$ on one side:

We achieve this by adding $12$ to both sides: $-2x - 12 + 12 > -17 + 12$.
This simplifies to $-2x > -5$.

By doing this, we move closer to isolating $x$, the variable of interest. This process helps to clarify what values $x$ can take by reducing the inequality to a simpler form.

Inequality Operations

Operations involving inequalities follow similar rules to equations, but with special consideration when multiplying or dividing by negative numbers. In the final step of solving $-2x > -5$, we divide both sides by $-2$:

Upon dividing by a negative number, remember to reverse the inequality sign.
This changes $-2x > -5$ to $x < \frac{5}{2}$ or $x < 2.5$.

This reversal is crucial in maintaining the inequality's validity.
Such operations ensure that our solution set properly describes all possible values for $x$ that satisfy the original inequality.
Without reversing the inequality, the solution would not accurately represent the relationship defined initially.

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91影视

For Problems \(1-50\), solve each inequality. (Objectives 1 and 2) $$ -2(x+6)>-17 $$

Short Answer

Step by step solution

Distribute the Negative Sign

Isolate Variable Term

Solve for x

Key Concepts

Distributive Property

Isolation of Variables

Inequality Operations

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