Two Absolute Values in One Problem

Let’s get started!

Solving for x

First tackle |2x – 3| = 9. Solve for x, using the Steps for Success for Absolute Values!

Step 1: Write the equation twice without the absolute value bars.

2x – 3 = 9             and        2x – 3 = 9

Step 2: With one of the equations, for everything that was inside the absolute value bars, change the sign to the opposite.

2x – 3 = 9             and        –2x + 3 = 9

Step 3: Solve each equation for the unknown.

We’ll start with 2x – 3 = 9

We have 2x = 12. Now let’s divide both sides by 2 to get x alone.

Now don’t forget the second equation of –2x + 3 = 9! We need to solve for the unknown with that one as well.

We have –2x = 6. Now let’s divide both sides by –2 to get x alone.

We have determined that x = 6 and x = –3

Step 4: Check your work! Substitute the answers you got back into the original equation.

Let’s check x = 6. We’ll substitute 6 in for x:

|2x – 3| = 9

|2(6) – 3| = 9

|12 – 3| = 9

|9| = 9

Correct!

Let’s check x = –3. We’ll substitute –3 in for x:

|2x – 3| = 9

|2(–3) – 3| = 9

|–6  – 3| = 9

|–9| = 9

Correct!

Solving for y

Now we need to go through this process to solve for y.

|3 – y| = 7

Step 1: Write the equation twice without the absolute value bars.

3 – y = 7               and        3 – y = 7

Step 2: With one of the equations, for everything that was inside the absolute value bars, change the sign to the opposite.

3 – y = 7               and        –3 + y = 7

Step 3: Solve each equation for the unknown.

We’ll start with 3 – y = 7

We have  – y = 4. Now let’s divide both sides by  –1  to get y alone.

Now don’t forget the second equation of –3 + y = 7! We need to solve for the unknown with that one as well.

y = 10

 We have determined that y = –4 and y = 10

Step 4: Check your work! Substitute the answers you got back into the original equation.

 Let’s check y = –4. We’ll substitute –4 in for y:

|3 – y| = 7

|3 – (–4)| = 7

|3 + 4| = 7

|7| = 7

Correct!

Let’s check y = 10. We’ll substitute 10 in for y:

|3 – y| = 7

|3 – 10| = 7

|–7| = 7

Correct!

We know that:

x = 6 or –3

y = –4 or 10

Let’s map out the different scenarios. First I hold the value for x the same and go through the different values of y (Scenarios 1 and 2). Then I go to the next value for x, and again go through the different values of y (Scenarios 3 and 4).

Let’s remind ourselves of what the question was:

We would chose Option a and Option c.

We wouldn’t choose Option b since the sign is wrong. We wouldn’t choose Option d since that didn’t match any of our solutions.

Great work!!