Question

Solve the inequality. Then graph the solution set on the real number line. \(\frac{x+6}{x+1}<2\)

Short answer

The solution to the inequality is \((-∞, -1) ∪ (4, ∞)\). These are the intervals on the real number line where the inequality is satisfied.

Step-by-step solution

Rearrange the Inequality

Subtract 2 from both sides of the inequality to isolate the fraction \(\frac{x+6}{x+1}\). The inequality should now be \(\frac{x+6}{x+1} - 2 <0\) which simplifies to \(\frac{x+6 - 2(x + 1)}{x+1}<0\), then to \(\frac{x+6-2x-2}{x+1}<0\), and ultimately \(\frac{-x+4}{x+1}<0\).

Solve for x

Next, find the values for x that make the numerator and the denominator of the inequality term equal to zero. The solutions are x = -1 from \(x + 1 = 0\) and x = 4 from \(-x + 4 = 0\). These values will represent critical points on the number line for the graph. Note that x cannot be -1 since that makes the denominator zero resulting in an undefined value.

Graph the Solution on the Real Number Line

On the number line, plot the points -1 and 4. Because of the less-than sign in the inequality, the solution will not include -1 and 4, so these points will be represented as open points on the number line. To determine the solution to the inequality, select test points to the left, between and to the right of your critical points on the number line, substitute them into the inequality and examine if it's true. For this inequality, a good choice of test points would be x = -2, x = 0, and x = 5. If the inequality is true, fill in the corresponding segment on the number line; if not, leave it empty.

Key concepts

Solving Inequalities

Inequalities appear similar to equations, but instead of an equal sign, they have symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). The goal is to find all possible values of the variable that make the inequality true. Let's look at solving the inequality \(\frac{x+6}{x+1}<2\).
First, isolate the variable expression by subtracting 2 from both sides, giving \(\frac{x+6}{x+1} - 2 < 0\). Simplify this to obtain \(\frac{-x+4}{x+1}<0\). These steps help analyze the inequality in a simpler form.
Next, find values of \(x\) that make the numerator and denominator zero. For the numerator \(-x + 4 = 0\), solve to find the critical point \(x = 4\). For the denominator \(x + 1 = 0\), find \(x = -1\). However, \(x = -1\) must be excluded from the solution since it makes the denominator zero, and results in an undefined expression.

Graphing on Number Line

Graphing inequalities on a number line involves identifying solution intervals. After finding critical points from the numerator and denominator, place these points on the number line.
In this example, there are critical points at \(x = -1\) and \(x = 4\). Mark these as open circles since neither point is included in the solution set, evidenced by the \(<\) symbol in the inequality. An open circle indicates that the number itself is not a valid solution.
To determine which intervals satisfy the inequality, choose test points in each interval created by your critical points. For instance, pick \(x = -2\) and \(x = 5\) in addition to the midpoint, \(x = 0\). Substitute each into the inequality \(\frac{-x+4}{x+1}<0\) to verify which intervals make it true. Graph accordingly by shading or marking the sections of the number line that satisfy the inequality.

Critical Points in Inequalities

Critical points are values where either the numerator or the denominator is zero. These points are vital in solving rational inequalities as they mark potential boundaries for solutions.
From our problem, \(x=4\) is a critical point because it satisfies \(-x+4=0\), making the rational expression zero. The critical point \(x=-1\) which makes \(x+1=0\) is noteworthy because it would make the expression undefined if mistakenly included. This makes the nature of critical points crucial to understanding and solving rational inequalities.
Remember:

• Solve the numerator and denominator separately to find critical points.
• Evaluate these points carefully for inclusion in the solution.
• Understanding where expressions change in concavity or identity is pivotal to graphing correct intervals on the number line.

Critical points act like hinge points that help balance and solve the inequality.