In application, one side of the inequality will often be a number (constant) instead of the variable “y.”

The solution set can be found using a graph or a table. Begin by graphing each side of the inequality on the same graph: y = x² + 7x + 6 (blue graph) and y = 0 (green graph).

x² + 7x + 6 ≥ 0 states that the left side (the quadratic or parabola) is greater than or equal to  the right side (y = 0).

The parabola divides the x-axis into three intervals (segments). In this case, those intervals are described by three linear inequalities:

Now determine the interval(s) of the axis which satisfy the conditions of the inequality.

We are looking for the interval where the function on the left side (x² + 7x + 6) is above (greater than) the function on the right (y = 0). Look at the points on the parabola that are above the green line on the graph. Which interval(s) of x values on the axis are below the parabola?

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The two intervals on the lower and upper ends

Therefore, the solution sets for this inequality are x ≤ -6 and x ≥ -1

If you change the inequality to x² + 7x + 6 ≤ 0, the solution set is now the middle interval of the x-axis or -6 ≤ x ≤ -1

On the table, look for the y values that satisfy the conditions of the inequality:

f(x) = x² + 7x + 6 ≤ 0

This means we look for y values that are 0 or negative. The blue shaded values on the table indicate those y-values.

The solution is the green shaded x-values (-6, -5, -4, -3, -2, -1) that generate the y-values (0, -4, -6, -6, -4 0), and all those in between the values listed on the table, since the function (and graph) is continuous.

Solution according to the table: -6 ≤ x ≤ -1 (same as derived from the graph).