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Chapter 2: Problem 28

Solve each inequality. Write the solution set in interval notation and graph it. $$ -\frac{7}{8} x \leq 21 $$

Short Answer

Expert verified
\([-24, \infty)\)

Step by step solution

01

Isolate the Variable

The inequality to solve is \(-\frac{7}{8}x \leq 21\). Start by isolating \(x\) by multiplying both sides by the reciprocal of \(-\frac{7}{8}\), which is \(-\frac{8}{7}\). Multiplying both sides by a negative number will flip the inequality sign, thus:\[ x \geq 21 \times \left(-\frac{8}{7}\right) \]
02

Simplify the Expression

Now, compute the right-hand side:\[ x \geq 21 \times \left(-\frac{8}{7}\right) = 21 \times -\frac{8}{7}\]Simplify:\[ x \geq -24 \]
03

Write the Solution in Interval Notation

The solution \(x \geq -24\) means that \(x\) can be any number greater than or equal to \(-24\). In interval notation, this solution is written as:\[ [-24, \infty) \]
04

Graph the Solution

On a number line, represent this solution by drawing a closed circle at \(-24\) and a line extending to the right towards infinity, indicating all values greater than or equal to \(-24\).

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Key Concepts

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

Solving Inequalities
Solving inequalities is a fundamental concept in algebra, similar to solving equations, but with a few additional rules. When you are given an inequality like \(-\frac{7}{8}x \leq 21\), the goal is to find all possible values of \(x\) that make the inequality true.
  • To solve inequalities, you follow similar steps as you would with equations: isolate the variable on one side.
  • However, a critical step in solving inequalities involves remembering that multiplying or dividing both sides of an inequality by a negative number flips the inequality sign.
  • This rule is essential because it preserves the truth of the inequality.
In the example exercise, to eliminate the fraction, you would multiply both sides of the inequality by the reciprocal of \(-\frac{7}{8}\), which is \(-\frac{8}{7}\). Doing so changes the inequality to \(x \geq -24\). Now, \(x\) is isolated, and the inequality is solved.
Interval Notation
Interval notation is a streamlined way to express the set of solutions for an inequality. It uses parentheses or brackets to show whether endpoints are included or excluded.
  • A bracket \([\) or \(]\) denotes that an endpoint is included in the solution set. For example, \([-24, \infty)\) means that -24 is part of the solution set.
  • A parenthesis \((\) or \()\) means the endpoint is not included. The infinity symbol always gets a parenthesis, as infinity isn't a number you can reach or include.
In this exercise, the solution \(x \geq -24\) means that \(x\) includes -24 and any number greater than it. Thus, it represents as \([-24, \infty)\) in interval notation. This concise representation is helpful in mathematics for expressing large sets of numbers easily.
Number Line Graphing
Number line graphing is a visual way to represent solutions of inequalities. For students and educators alike, it's a great method to understand and convey the set of possible solution values.
  • To graph the solution \(x \geq -24\) on a number line, place a closed circle on -24 to indicate it鈥檚 included in the solution set.
  • Draw a line extending to the right side of the number line, toward infinity, showing that all numbers greater than -24 are included.
  • The rightward direction signifies that the solution set contains all numbers larger than -24, matching the inequality \(x \geq -24\).
This visual representation helps confirm that the solution includes and extends beyond the number presented on the number line, reinforcing the interval notation \([-24, \infty)\) used earlier.

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Most popular questions from this chapter

Solve each equation and inequality. Write the solution set of each inequality in interval notation and graph it. $$ \text { a. }-16<4(x+8) \leq 8 \quad \text { b. }-16=4(x+8)+8 $$

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ y-\frac{1}{7} \leq \frac{2}{3} $$

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 7<\frac{5}{3} a+(-3) $$

Dry mixture problems. A store sells bluegrass seed for \(\$ 6\) per pound and ryegrass seed for 3 dollar per pound. How much ryegrass must be mixed with 100 pounds of bluegrass to obtain a blend that will sell for 5 dollar per pound?

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 7 x-16<2 x+4 x $$
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