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Chapter 1: Problem 22

Solve each inequality. Write the solution set using interval notation and graph it. $$\frac{5-x}{3} \leq-2$$

Short Answer

Expert verified
[11, \infty)

Step by step solution

01

- Eliminate the Fraction

To begin solving the inequality \(\frac{5 - x}{3} \leq -2\), multiply both sides of the inequality by 3 to eliminate the fraction. This gives: \[5 - x \leq -6\].
02

- Isolate the Variable

Next, isolate the variable \(x\). Subtract 5 from both sides of the inequality to get: \[ -x \leq -11 \].
03

- Solve for x

Since the coefficient of \(x\) is negative, multiply both sides by -1 and reverse the inequality sign. This results in: \[ x \geq 11 \].
04

- Write Solution in Interval Notation

Now, write the solution in interval notation. The solution \(x \geq 11\) corresponds to the interval \[ [11, \infty) \].
05

- Graph the Solution

To graph the solution on a number line, draw a closed circle at 11 and shade the line to the right, indicating all numbers greater than or equal to 11.

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Key Concepts

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

interval notation
Interval notation makes it easier to express ranges of values in a compact form. It's especially useful in inequalities. Here, you represent the solution with specific symbols:
  • A square bracket \[ \]\ indicates that the endpoint is included (closed interval).
  • A parenthesis \( \)\ indicates that the endpoint is not included (open interval).
For example, when you solve the inequality \[ x \geq 11 \], you're looking for all numbers 11 and above. Using interval notation, you can write this as \[ [11, \infty) \]. The square bracket means 11 is included, and the parenthesis means infinity is never truly reached, so it's not included. Understanding this compact form helps you quickly see the range of solutions.
isolating variables
Isolating the variable is a key step in solving inequalities. It involves manipulating the equation to get the variable by itself on one side of the inequality. Here's how it works step-by-step:
  • Look at the inequality \[ \frac{5 - x}{3} \leq -2 \]. The first step involves getting rid of the fraction by multiplying both sides by 3, resulting in \[ 5 - x \leq -6 \].
  • Next, you need to get rid of the constant on the same side as the variable. Subtract 5 from both sides to get \[ -x \leq -11 \].
  • Finally, since \( x \) is negative, multiply both sides by -1 and remember to reverse the inequality sign. This gives you \[ x \geq 11 \].
Now the variable is isolated, and the inequality is much simpler. Practice this method to become more comfortable with these steps.
graphing inequalities
Graphing inequalities helps you visualize the range of solutions and understand the concept better. Here's how you can graph the solution for \[ x \geq 11 \]:
  • Start by drawing a number line.
  • Locate the point 11 on this line.
  • Since the inequality is \[ x \geq 11 \], put a closed circle at 11 because 11 itself is included in the solution set.
  • Shade the line to the right of 11 to show all numbers greater than or equal to 11.
This visual aid makes it easier to understand the solution set of the inequality. Remember, a closed circle means the number is included, and an open circle means it is not included. Practice graphing different inequalities to strengthen your understanding.

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