In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y1=x2+3,y2=x3+52,andy1≤y2y_1 = \frac{x}{2} + 3, \quad $$y_2$$ = \frac{x}{3} + \frac{5}{2}, \quad \text{and} \quad $$y_1$$ \leq $$y_2 y1$$=2x+3,y2=3x+25,andy1≤y2

Ch. 1 - Equations and Inequalities

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Ch. 1 - Equations and Inequalities

문제 95

2장, 문제 95

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $y_1 = $\frac{x}{2}$ + 3, $\quad$ y_2 = $\frac{x}{3}$ + $\frac{5}{2}$, $\quad$ $\text{and}$ $\quad$ y_1 $\leq$ y_2$

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Start with the inequality given: $y_1 \leq y_2$, which means $\frac{x}{2} + 3 \leq \frac{x}{3} + \frac{5}{2}$.

To solve for $x$, first eliminate the fractions by finding a common denominator or multiply both sides by the least common multiple of the denominators (which is 6) to clear the fractions.

After clearing fractions, simplify the inequality to isolate terms involving $x$ on one side and constants on the other side.

Solve the resulting linear inequality for $x$ by performing algebraic operations such as addition, subtraction, multiplication, or division, keeping in mind to reverse the inequality sign if multiplying or dividing by a negative number.

Express the solution set for $x$ using interval notation, which describes all values of $x$ that satisfy the inequality.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inequalities

Inequalities express a relationship where one quantity is less than or equal to another. Solving inequalities involves finding all values of the variable that satisfy this condition, often resulting in a range or interval of solutions rather than a single value.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses brackets [ ] for inclusive boundaries and parentheses ( ) for exclusive boundaries, making it ideal for expressing solution sets of inequalities.

Linear Functions and Graphs

Linear functions have the form y = mx + b, where m is the slope and b is the y-intercept. Understanding how to compare two linear functions helps determine where one is less than or equal to the other, which is essential for solving inequalities involving linear expressions.

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y1=x2+3,y2=x3+52,andy1≤y2y_1 = \(\frac{x}{2}\) + 3, \(\quad\) y_2 = \(\frac{x}{3}\) + \(\frac{5}{2}\), \(\quad\) \(\text{and}\) \(\quad\) y_1 \(\leq\) y_2 y1=2x+3,y2=3x+25,andy1≤y2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inequalities

Interval Notation

Linear Functions and Graphs

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $y_1 = \(\frac{x}{2}\) + 3, \(\quad\) y_2 = \(\frac{x}{3}\) + \(\frac{5}{2}\), \(\quad\) \(\text{and}\) \(\quad\) y_1 \(\leq\) y_2$