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Chapter 1: Problem 14

Solve each inequality. Write each solution set in interval notation. $$-3 x-8 \leq 7$$

Short Answer

Expert verified
The solution is

Step by step solution

01

Isolate the x term

To isolate the term with the variable, add 8 to both sides of the inequality:
02

Simplify the inequality

After adding 8 to both sides, the inequality becomes:
03

Divide by -3

Now, divide both sides of the inequality by -3. Remember, dividing by a negative number reverses the inequality sign:
04

Write in interval notation

The solution in interval notation is:

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Key Concepts

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

Isolation of the Inequality
To solve an inequality, you first need to isolate the variable (in this case, let's focus on isolating \(x\)). Let's look at the inequality:
\[-3x - 8 \leq 7\]
Start by eliminating any constants on the side with the variable. Here, we'll add 8 to both sides:
\[-3x - 8 + 8 \leq 7 + 8\]
This simplifies to:
\[-3x \leq 15\]
Now, \(x\) is isolated with a coefficient of -3.
Reverse Inequality Sign
Handling inequalities involves a crucial rule: when you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. After isolating the term with \(x\), our inequality is:
\[-3x \leq 15\]
As the next step, we divide both sides by -3. Here's the key part鈥攂ecause we are dividing by a negative number, the inequality sign will flip:
\[x \geq -5\]
This step is essential for correctly solving inequalities. Always remember to reverse the inequality sign when dealing with negative multipliers or divisors.
Interval Notation
The final part of solving an inequality involves writing the solution in interval notation. This is a concise way to represent all possible solutions. From our final inequality \(x \geq -5\), it means \(x\) can be any number greater than or equal to -5.
In interval notation, this is written as:
\[[-5, \infty)\]
Here鈥檚 how to break it down:
  • The bracket [ means -5 is included in the solution (\(x\) can be equal to -5).
  • The parenthesis ) after infinity means infinity is not a specific number we can reach, so we use ( for open intervals.
This notation compactly expresses that \(x\) can be any number from -5 to infinity.

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Most popular questions from this chapter

Write each statement as an absolute value equation or inequality. \(m\) is no more than 2 units from 7.

Find each product. Write the answer in standard form. $$-5 i(4-3 i)^{2}$$

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{1-3 i}{1+i}$$

Solve each equation or inequality. $$|4 x+3|>0$$

When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate \(R_{L}\) (in grams per hour) during a lecture class satisfied the inequality $$\left|R_{L}-26.75\right| \leq 1.42,$$ whereas during an exam the rate \(R_{E}\) satisfied the inequality $$\left|R_{E}-38.75\right| \leq 2.17.$$ Use this information in Exercises. The class had 225 students. If \(T_{L}\) and \(T_{E}\) represent the total amounts of carbon dioxide in grams emitted during a one-hour lecture and exam, respectively, write inequalities that model the ranges for \(T_{L}\) and \(T_{E}\).
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