Chapter 1: Problem 18
Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{7}{4 x} \leq 7 $$
Short Answer
Expert verified
The solution is \( x \in \left[ \frac{1}{4}, \infty \right) \).
Step by step solution
01
Clear the Fraction
Firstly, we need to get rid of the fraction in the inequality \( \frac{7}{4x} \leq 7 \). To do this, multiply both sides of the inequality by \( 4x \) (assuming \( x eq 0 \)) to clear the fraction. This gives us:\[ 7 \leq 28x \]
02
Simplify the Equation
Next, we simplify the inequality from Step 1. First, divide all parts of the inequality by 28 to solve for \( x \):\[ x \geq \frac{1}{4} \]
03
Express the Solution in Interval Notation
We now express the range of the solution in interval notation. From Step 2, we know \( x \) must be greater than or equal to \( \frac{1}{4} \). This translates to:\[ x \in \left[ \frac{1}{4}, \infty \right) \]
04
Sketch the Graph
To sketch the graph of the solution set, draw a number line. Mark the point \( \frac{1}{4} \) and shade the region to the right of this point, including \( \frac{1}{4} \), because \( x \) is greater than or equal to \( \frac{1}{4} \). Use a solid dot on \( \frac{1}{4} \) to indicate that it is included in the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
When solving inequalities, expressing the solution set in interval notation helps clearly communicate the range of values that satisfy the inequality. In interval notation, we use brackets to show which end points are included or excluded in the set.
- A square bracket "\[" or "\]" indicates that the endpoint is included in the set. This means the value satisfies the inequality, such as when we have "greater than or equal to" (≥) or "less than or equal to" (≤).- A parenthesis "(" or ")" indicates that the value is excluded, as with "greater than" (>) or "less than" (<).In the given solution, the expression \( x \in \left[ \frac{1}{4}, \infty \right) \) shows that \( x \) starts from \( \frac{1}{4} \) and goes on to infinity, including \( \frac{1}{4} \). The left end of the interval uses a square bracket \([\) because \( \frac{1}{4} \) is a valid solution, while infinity always uses a parenthesis ")" because it is a concept rather than a number.
- A square bracket "\[" or "\]" indicates that the endpoint is included in the set. This means the value satisfies the inequality, such as when we have "greater than or equal to" (≥) or "less than or equal to" (≤).- A parenthesis "(" or ")" indicates that the value is excluded, as with "greater than" (>) or "less than" (<).In the given solution, the expression \( x \in \left[ \frac{1}{4}, \infty \right) \) shows that \( x \) starts from \( \frac{1}{4} \) and goes on to infinity, including \( \frac{1}{4} \). The left end of the interval uses a square bracket \([\) because \( \frac{1}{4} \) is a valid solution, while infinity always uses a parenthesis ")" because it is a concept rather than a number.
Number Line Graphing
Number line graphing is a visual way to depict the range of solutions for an inequality. This method provides a clear representation of which numbers are solutions and helps to visualize how the solution set behaves.
To graph the solution \( x \geq \frac{1}{4} \), start by drawing a horizontal line. - Mark the point \( \frac{1}{4} \) on this line. This point represents the threshold beyond which all numbers are solutions to the inequality.- Since the inequality is "greater than or equal to", this point is included in the solution set. Use a solid dot or circle at \( \frac{1}{4} \).- Draw a shaded line or arrow extending to the right from \( \frac{1}{4} \) to indicate all numbers greater than \( \frac{1}{4} \) are included.This method provides an intuitive grasp of the solution, showing that any number equal to or larger than \( \frac{1}{4} \) is valid, which aligns with the interval notation \( \left[ \frac{1}{4}, \infty \right) \).
To graph the solution \( x \geq \frac{1}{4} \), start by drawing a horizontal line. - Mark the point \( \frac{1}{4} \) on this line. This point represents the threshold beyond which all numbers are solutions to the inequality.- Since the inequality is "greater than or equal to", this point is included in the solution set. Use a solid dot or circle at \( \frac{1}{4} \).- Draw a shaded line or arrow extending to the right from \( \frac{1}{4} \) to indicate all numbers greater than \( \frac{1}{4} \) are included.This method provides an intuitive grasp of the solution, showing that any number equal to or larger than \( \frac{1}{4} \) is valid, which aligns with the interval notation \( \left[ \frac{1}{4}, \infty \right) \).
Fraction Manipulation
Fraction manipulation is a crucial step in solving many inequalities and equations, especially those involving terms in a denominator. The primary aim is to clear the fractions to simplify the inequality into a more easily solvable form.
In the inequality \( \frac{7}{4x} \leq 7 \), the fraction \( \frac{7}{4x} \) forms a hurdle. To eliminate it, multiply both sides by the denominator term (in this case, \( 4x \)) which balances the equation away from fractional terms.- It is essential to assume \( x eq 0 \) because division by zero is undefined.- After multiplying through, the inequality simplifies to \( 7 \leq 28x \), which is free of fractions and easier to handle.Mastering fraction manipulation can greatly streamline the process of solving inequalities, turning complex equations into simple expressions. Br>
In the inequality \( \frac{7}{4x} \leq 7 \), the fraction \( \frac{7}{4x} \) forms a hurdle. To eliminate it, multiply both sides by the denominator term (in this case, \( 4x \)) which balances the equation away from fractional terms.- It is essential to assume \( x eq 0 \) because division by zero is undefined.- After multiplying through, the inequality simplifies to \( 7 \leq 28x \), which is free of fractions and easier to handle.Mastering fraction manipulation can greatly streamline the process of solving inequalities, turning complex equations into simple expressions. Br>
Solving Inequalities
Solving inequalities involves determining the range of values that satisfy the given logical statement. It is similar to solving equations, but one must account for order relations (greater than, less than, etc.).
To solve \( 7 \leq 28x \), perform operations that isolate \( x \) on one side of the inequality:- Divide both sides by 28 to solve for \( x \). Here, the result is \( x \geq \frac{1}{4} \). It's crucial to remember the flip rule: if you multiply or divide both sides by a negative number, reverse the inequality sign.Once solved, the solution set can be further expressed by other means such as interval notation and graphing, enhancing understanding and communication of solutions. Becoming proficient in solving inequalities is a fundamental skill, especially helpful in fields like mathematics, economics, and engineering.
To solve \( 7 \leq 28x \), perform operations that isolate \( x \) on one side of the inequality:- Divide both sides by 28 to solve for \( x \). Here, the result is \( x \geq \frac{1}{4} \). It's crucial to remember the flip rule: if you multiply or divide both sides by a negative number, reverse the inequality sign.Once solved, the solution set can be further expressed by other means such as interval notation and graphing, enhancing understanding and communication of solutions. Becoming proficient in solving inequalities is a fundamental skill, especially helpful in fields like mathematics, economics, and engineering.
