Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 6x²+x>1

Verified step by step guidance

1. Rewrite the inequality so that one side is zero: $\mathrm{6x^2}+x-1>0$ becomes $\mathrm{6x^2}+x-1>0$.
2. Factor the quadratic expression on the left side, if possible. To do this, look for two numbers that multiply to $6\; imes\; (-1)\; =\; -6$ and add to $1$. Then rewrite the middle term and factor by grouping.
3. Find the roots of the quadratic equation $6x^2\; +\; x\; -\; 1\; =\; 0$ by using the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$, where $a=6$, $b=1$, and $c=-1$.
4. Use the roots to divide the real number line into intervals. Test a value from each interval in the inequality $6x^2\; +\; x\; -\; 1\; >\; 0$ to determine where the inequality holds true.
5. Express the solution set as an interval or union of intervals based on the test results, and then graph this solution set on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols like >, <, ≥, or ≤. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring and Finding Critical Points

To solve polynomial inequalities, first rewrite the inequality so one side is zero, then factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested to determine where the inequality holds.

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality is true, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a real number line visually represents these intervals, using open or closed dots to indicate whether endpoints are included.