Question

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 	ext{and} \quad $$y_1$$ \leq $$y_2
y1$$​=2x​+3,y2​=3x​+25​,andy1​≤y2​

Accepted Answer

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.

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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Key Concepts

Inequalities

Interval Notation

Linear Functions and Graphs

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_2y1=2x+3,y2=3x+25,andy1≤y2