Inequalities Explained Translating, Solving, And Graphing

When dealing with mathematical inequalities, it's crucial to accurately translate real-world scenarios into symbolic representations. In this case, we're presented with the statement: "The discount is good on any purchase under $20." The key phrase here is "under", which indicates that the discount applies to purchases less than $20. Inequalities are mathematical expressions that show the relationship between two values that are not necessarily equal. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). To accurately represent the given statement, we need to choose the inequality symbol that corresponds to "under," which implies a value strictly less than $20.

Let's break down the options provided:

• A. $d > 20$: This inequality states that the discount ($d$) applies to purchases greater than $20. This is the opposite of what the original statement conveys.
• B. $d < 20$: This inequality states that the discount ($d$) applies to purchases less than $20. This aligns perfectly with the statement "the discount is good on any purchase under $20."
• C. $d ≤ 20$: This inequality states that the discount ($d$) applies to purchases less than or equal to $20. While it includes values less than $20, it also includes $20 itself. The original statement specifically says "under $20," implying that $20 is not included.
• D. $d ≥ 20$: This inequality states that the discount ($d$) applies to purchases greater than or equal to $20. This is the opposite of what the original statement conveys and includes $20, which should be excluded.

Therefore, the correct inequality to represent the given sentence is B. $d < 20$. This inequality accurately captures the condition that the discount is valid for purchases strictly less than $20. Understanding the nuances of inequality symbols is essential for correctly translating real-world scenarios into mathematical expressions and for solving a wide range of problems in algebra and beyond. This ability to translate words into mathematical symbols is a fundamental skill in mathematics, applicable not only to inequalities but also to equations, functions, and various other mathematical concepts. Mastering this skill is crucial for success in higher-level mathematics and in practical applications where mathematical modeling is required.

Solving linear inequalities is a fundamental skill in algebra, with applications in various fields such as optimization, economics, and computer science. The process is quite similar to solving linear equations, but with one important difference: multiplying or dividing by a negative number requires flipping the inequality sign. This rule is crucial for maintaining the accuracy of the solution. In this problem, we're tasked with solving the inequality $-15

To solve this inequality, our goal is to isolate the variable f on one side. We can achieve this by performing operations on both sides of the inequality, maintaining the balance, just as we do with equations. The key operation here is to add 21 to both sides of the inequality. This will eliminate the -21 term on the right side, leaving us with f isolated:

$-15

$-15 + 21

$6

Now, we have the inequality $6

• A. $f ≥ 6$: This inequality states that f is greater than or equal to 6. This is the opposite of what our solution indicates.
• B. $f ≤ -6$: This inequality states that f is less than or equal to -6. This is incorrect, as our solution clearly shows that f should be less than or equal to 6, not -6.
• C. $f ≤ 6$: This inequality states that f is less than or equal to 6. This perfectly matches our derived solution.
• D. $f ≥ -6$: This inequality states that f is greater than or equal to -6. This is incorrect and does not align with our solution.

Therefore, the correct solution to the inequality $-15

In summary, solving linear inequalities involves similar steps to solving linear equations, but with the crucial additional rule of flipping the inequality sign when multiplying or dividing by a negative number. This skill is essential for a strong foundation in algebra and for tackling more complex mathematical problems.

Graphing inequalities is a crucial skill in algebra as it provides a visual representation of the solution set. Unlike equations which typically have a single solution or a finite set of solutions, inequalities often have an infinite range of solutions. Graphing these solutions on a number line or a coordinate plane helps to understand the range of values that satisfy the inequality. The method for graphing depends on whether we are dealing with a single-variable inequality (graphed on a number line) or a two-variable inequality (graphed on a coordinate plane).

When graphing inequalities on a number line, we use two primary visual cues: open circles and closed circles. An open circle is used to indicate that the endpoint of the interval is not included in the solution set. This occurs when the inequality uses a "less than" (<) or "greater than" (>) symbol. For instance, if we have the inequality $x > 3$, we would place an open circle at 3 on the number line, indicating that 3 itself is not a solution, but all numbers greater than 3 are. Conversely, a closed circle is used to indicate that the endpoint of the interval is included in the solution set. This is used when the inequality uses a "less than or equal to" (≤) or "greater than or equal to" (≥) symbol. For example, if we have the inequality $x ≤ 5$, we would place a closed circle at 5 on the number line, meaning that 5 is a solution, along with all numbers less than 5.

After placing the circle, we then shade the number line in the direction that represents the solutions. For a "greater than" (>) or "greater than or equal to" (≥) inequality, we shade to the right, indicating that all numbers to the right of the endpoint are solutions. For a "less than" (<) or "less than or equal to" (≤) inequality, we shade to the left, indicating that all numbers to the left of the endpoint are solutions. For two-variable inequalities, which are graphed on a coordinate plane, we use a similar approach, but instead of circles, we use dashed or solid lines and shade regions of the plane. A dashed line represents that the points on the line are not included in the solution, which corresponds to inequalities with < or > symbols. A solid line indicates that the points on the line are included in the solution, which corresponds to inequalities with ≤ or ≥ symbols. After drawing the line, we shade the region above or below the line, depending on the inequality. For an inequality in the form $y > mx + b$ or $y ≥ mx + b$, we shade above the line, while for $y < mx + b$ or $y ≤ mx + b$, we shade below the line. The shaded region represents all the points (x, y) that satisfy the inequality.

Graphing inequalities provides a powerful visual tool for understanding solutions that span a range of values. Whether on a number line or a coordinate plane, the principles of using open and closed circles (or dashed and solid lines) and shading the appropriate region are fundamental for accurately representing the solution set of an inequality.