The solutions to the inequality are all values of x less than or equal to 4. We should draw a number line and mark 4 as a closed point, since the inequality is non-strict.
To complete the graph, the part of the line that is less than 4 should be shaded.
b To determine the solution set we will first have to find its boundary point(s) by treating the inequality as an equation.
Inequality:& x^2+6>42
Equation:& x^2+6=42
If we solve this equation, we can determine the boundary points.
The boundary points are located at x=-6 and x=6. Let's mark them on a number line. Since the inequality is strict, the boundaries are not included in the solution set. To determine where we should shade the number line, we will also include three test points.
Now, substitute x in the inequality with these values and see which one holds.
|c|c|c|
[-0.8em]
x & x^2+6>42 & Evaluate [0.5em]
[-1em]
-8 & ( -8)^2+6>42 & 70 > 42 ✓ [0.5em]
[-0.7em]
0 & ( 0)^2+6>42 & 6 ≯ 42 * [0.5em]
[-1em]
8 & ( 8)^2+6>42 & 70 > 42 ✓ [0.5em]
The inequality is true to the left and right of the boundary points. Therefore, we should shade these regions.
