Question

|2x + 5| < 13 how do I solve and graph this absolute value inequality

Answer 1

Answer:

  -9 < x < 4

  see the attachment for a graph

Step-by-step explanation:

An absolute value expression resolves into a piecewise-defined expression. Equations involving them are solved, in general, by solving the equation in each of the domains of the piecewise-defined function(s).

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general case

  y = |a| means: y = a, for a ≥ 0; and y = -a, for a < 0

When this is applied to an equation or inequality, you get ...

  |2x +5| < 13 means:

• 2x +5 < 13   for (2x+5) ≥ 0
• -(2x +5) < 13   for (2x +5) < 0

Solving these inequalities and merging the solutions is generally how you go about working a problem like this.

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simpler case

However, when the inequality is in the form you have here:

  |a| < b

this whole process comes down to solving the compound inequality ...

  -b < a < b

  -13 < 2x +5 < 13 . . . . . applied to your inequality

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solution

This is solved using the same steps you would use for a 2-step linear equation:

  -18 < 2x < 8 . . . . . . . subtract the constant 5

  -9 < x < 4 . . . . . . . . divide by the coefficient of x

The solution to the inequality is -9 < x < 4.

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graph

A one-variable inequality is graphed on a number line.

The < symbols mean the end values (-9 and 4) are not part of the solution set, so open circles are drawn at those points on the number line. The inequality tells you that values between -9 and 4 are in the solution set, so the number line is shaded there.

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Additional comment

One of the ways that |a| < b inequalities can be solved is to graph y1=|a| and y2=b and look for the place(s) on the graph where y1 < y2. This sort of approach is shown in the second attachment. The red graph (partially covered by the dashed green line) is the absolute value function. The upper blue line is y=13, so the portion of the red graph below that line is where |2x+5|<13. The orange shading shows that region.

The purpose of the green dashed line and the bottom blue line is to show the equivalent solution we used here. Effectively, we reflected the problem across the x-axis for the domain where (2x+5) < 0. You notice that the values of x that satisfy -13 < 2x+5 < 13 are the same values of x that satisfy |2x+5| < 13.

advanced idea

This same "reflect the problem across the x-axis" idea can be used for inequalities of the form |a| > b. However, writing this as (-b > a > b) is problematic. It needs to be written as the system {-b > a, a > b} whose solution is the union of the disjoint parts of the solution set.