For Exercises 31-34: (a) Use a graph [as in Example 7(a)] to...

For Exercises 31-34: (a) Use a graph [as in Example 7(a)] to determine which of the following general forms describes the solution set of the given inequality: $$ [a, b] \quad(-\infty, a) \cup(b, \infty) \quad[a, \infty) \quad(a, \infty) $$ (b) Use a graphing utility [as in Example 7(b)] to estimate to the nearest hundredth the value of $a$, and where appropriate, $b$. (c) Solve the inequality algebraically and write the solution set using interval notation. Check that your answers are consistent with the graphical results in parts (a) and (b). $7 x-2 \geq 0$

For Exercises 31-34:
(a) Use a graph [as in Example 7(a)] to determine which of the following general forms describes the solution set of the given inequality:
$$
[a, b] \quad(-\infty, a) \cup(b, \infty) \quad[a, \infty) \quad(a, \infty)
$$
(b) Use a graphing utility [as in Example 7(b)] to estimate to the nearest hundredth the value of $a$, and where appropriate, $b$.
(c) Solve the inequality algebraically and write the solution set using interval notation. Check that your answers are consistent with the graphical results in parts (a) and (b).
$7 x-2 \geq 0$

Key Concepts

Linear Inequalities

Linear inequalities involve expressions of the first degree and require finding all values for the variable that make the inequality true. The process typically involves isolating the variable by performing algebraic operations similar to those used in solving equations, while being careful to reverse the inequality sign when multiplying or dividing by a negative number.

Graphical Representation

Graphically representing an inequality involves plotting the boundary line or point corresponding to the equality (where the expression equals zero) and then shading the regions on the number line or coordinate plane where the inequality holds. A closed circle indicates that the endpoint is included (using a bracket in interval notation), while an open circle indicates exclusion.

Interval Notation

Interval notation is a concise method for writing subsets of the real numbers that constitute the solution sets of inequalities. It uses brackets to denote inclusion of endpoints and parentheses to denote exclusion, efficiently conveying which portions of the number line satisfy the inequality.

Graphing Utilities for Estimation

Graphing utilities help by visually displaying the function associated with the inequality, making it easier to approximate the solution, especially the boundary points. These tools can confirm the algebraic solutions by providing numerical estimates and graphs that show where the function meets or exceeds the required value.

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