SOLVING ABSOLUTE VALUE EQUATIONS GRAPHICALLY

Solve for x using

i) graphical method    ii) an algebraical method.

Problem 1 :

|x + 2| = 2x + 1

Solution :

Graphing absolute value equation :

Let y = |x + 2| = 2x + 1

y = |x + 2| and y = 2x + 1

Comparing y = |x + 2| with y = a|x - h| + k

(h, k) is (-2, 0). a = 1, the curve is opening up.

By applying x = 0, we get y = 2. So, (0, 2) is the y-intercept.

Graphing the line :

y = 2x + 1

Slope (m) = 2

It is a raising line.

Tracing the next point after y-intercept of absolute value function, we land at (1, 3).

Using slope, we can find the next point on the line. From 1 on the y-axis move 2 units above and 1 unit to the right. We will be landing on the point (1, 3). So, the solution is (1, 3).

Solving algebraically :

|x + 2| = 2x + 1

Considering the x - intercepts of the line and absolute value function, we should take the value of x as 1.

Applying the value of x in the original function, we will get y = 3.So, (1, 3) is the point of intersection.

Problem 2 :

Consider the graph, which shows y₁ = |6 - 2x| and y₂ = 18. Use the graph and use algebraic properties to solve |6 - 2x| = 18.

Solution :

|6 - 2x| = 18

Decomposing into two branches, we get

By observing the graph, the point of intersections are -6 and 12.

Problem 3 :

You are asked for the year of the Emancipation Proclamation in the United States on a test. The correct answer is 1863. You guessed g and you were off by 4 years. What equation’s solution gives the possible values of g?

a) |1863 - 4| = g      b) |g| = 1863 - 4     c) |g - 1863|  = 4    d) |g - 4|  = 1863

Solution :

• The target value (actual year) is 1863
• The guess is g
• The "distance" or error between the guess and the target is 4 years.

To find the possible values of g, we set the distance between and equal to :

|g - 1863|  = 4

Problem 4 :

Determine whether the number is a solution to the equation

60 = |n - 90| 

a) 30     b) −30    c) 150     d) −150

Solution :

60 = |n - 90| 

So, 30 and 150 are solutions.

Problem 5 :

 Use the table below to solve each sentence.

a) |2x - 3| = 7     b) |2x - 3| < 7     c) |2x - 3| > 7

Solution :

Option a :

|2x - 3| = 7

Option b :

|2x - 3| < 7

-2 < x < 5

Option c :

|2x - 3| > 7

(-∞, -2) (5, ∞)