Problem 40 $6 x+7 \leq-17$... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 2: Problem 40

$6 x+7 \leq-17$

Short Answer

Expert verified

x \leq -4

Step by step solution

- Isolate the Variable Term

Start by isolating the term containing the variable on one side of the inequality. Subtract 7 from both sides of the inequality: \[ 6x + 7 - 7 \leq -17 - 7 \] This simplifies to: \[ 6x \leq -24 \]

- Solve for the Variable

Next, solve for the variable by dividing both sides by 6: \[ \frac{6x}{6} \leq \frac{-24}{6} \] This simplifies to: \[ x \leq -4 \]

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Key Concepts

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

isolate the variable

When solving inequalities like this, our first goal is to isolate the variable term. That means we want to get the term with the variable by itself on one side of the inequality. In the given problem, the variable is within the term `6x`. To isolate `6x`, we subtract 7 from both sides. Imagine you're balancing a scale: whatever you do to one side, you have to do to the other. By subtracting 7 from both sides, we ensure the balance is maintained.
So the equation changes from: <
To:

This is the first step to `isolate` the variable, setting the stage for the next operation.

algebraic manipulation

Algebraic manipulation refers to the techniques we use to solve for the variable. In this exercise, after isolating the variable term, we need to solve for `x` by itself. This is done by dividing both sides of the inequality by 6.
Remember, division here is evenly spreading the value, much like splitting a pie among friends. Dividing both sides by 6 simplifies the equation to `x 鈮� -4`. These steps show two key manipulations: subtraction and division. Using these fundamental moves correctly is crucial in algebra.

linear inequalities

Linear inequalities resemble linear equations, but they instead use inequality signs like 鈮�, 鈮�, <, or >. They show a range of values rather than one specific number. In this case, `6x + 7 鈮� -17`, it's a linear inequality because it involves `x` and maintains a linear relationship.
The same rules apply for solving these as for linear equations, with extra attention to the inequality sign. For instance, if we multiply or divide both sides by a negative number, we must flip the inequality sign.

mathematics education

Understanding how to solve inequalities is a key part of mathematics education. It builds on the concepts of equations while introducing the idea of ranges of solutions. Mastering these skills helps students apply logical thinking and problem-solving in real-world situations.
Approaching problems step-by-step and practicing these techniques helps reinforce learning. Algebra is foundational for advanced math and many scientific fields, illustrating its importance in a well-rounded education.

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91影视

\(6 x+7 \leq-17\)

Short Answer

Step by step solution

- Isolate the Variable Term

- Solve for the Variable

Key Concepts

isolate the variable

algebraic manipulation

linear inequalities

mathematics education

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