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Big Ideas Math Algebra 1, 2015
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Big Ideas Math Algebra 1, 2015 View details
Cumulative Assessment
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Exercise 8 Page 49

For the absolute value equations, rewrite them in the form |ax+b|=c. For the linear equations, simplify the left-hand side and right-hand side of the equations as much as possible.

No solution One solution Two solutions Infinitely many solutions
0=|x+13|+2, 3x-12=3(x-4)+1 |8x+3|=0, 12x-2=10x-8, - 2x+4=2x+4, - 6=5x-9 9=3|2x-11| - 4(x+4)=- 4x-16, 7-2x=3-2(x-2)
Practice makes perfect

We have multiple absolute value equations and linear equations to consider. Let's do it one at a time.

Absolute Value Equation

An absolute value equation can have 0, 1, or 2 solutions.

  • solutions: The absolute value of an expression is always non-negative. Therefore, whenever you have an absolute value equation that equals a negative number, it will not have a solution.
  • 1 solution: Absolute value equations represent distance from a midpoint on a number line. If that distance is 0, then there is only 1 solution.
  • 2 solutions: If the distance is greater than 0 there are 2 solutions, each on either side of the midpoint.
Out of the nine equations, three of them are absolute value equations. Before we can classify them, we have to make sure they are all written in the form |ax+b|=c.

0&=|x+13|+2 &&⇔ |x+13|=- 2 9&=3|2x-11| &&⇔ |2x-11|= 3 Now we can classify each of the absolute value equations.

Equation Solutions
|8x+3|=0 1
|x+13|=- 2 0
|2x-11|= 3 2

Linear Equation

A linear equation can have 0, 1, or infinite solutions.

  • solutions: Both sides of a linear equation can be represented as graphs of a line in a coordinate plane. When the graphs of two linear equations are parallel, they never intersect, which means there is no solution.
  • 1 solution: When the graphs of two linear expressions have different slopes, there is 1 solution.
  • Infinite solutions: When the graphs of two linear expressions are identical, they overlap and therefore have infinitely many solutions.

To classify the linear equations, we first have to simplify the left-hand and right-hand side in the equations where we can. 3x-12&=3(x-4)+1 &&⇔ 3x-12=3x-11 - 4(x+4)&=- 4x-16 &&⇔ - 4x-16=- 4x-16 7-2x&=3-2(x-2) &&⇔ 7-2x=7-2x Now we can determine how many solutions the equations have.

Equation Solutions
- 2x+4=2x+4 1
12x-2=10x-8 1
- 6=5x-9 1
3x-12=3x-11 0
- 4x-16=- 4x-16 Infinite
7-2x=7-2x Infinite

See that in the first equation the slopes - 2 and 2 are different, which is why there is one solution. In the fourth equation the slopes are the same and equal to 3, but the constant terms are different. 3x-12= 3x-11 This means that the lines are parallel and there are no solutions. In the fifth equation both sides are equal, which means there are infinitely many solutions. Now we can complete the table from the exercise.

No solution One solution Two solutions Infinitely many solutions
0=|x+13|+2, 3x-12=3(x-4)+1 |8x+3|=0, 12x-2=10x-8, - 2x+4=2x+4, - 6=5x-9 9=3|2x-11| - 4(x+4)=- 4x-16, 7-2x=3-2(x-2)
Subchapter links expand_more
arrow_right Exercises p.48-49
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Exercises 3 (Page 48)
Exercises 4 (Page 48)
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Exercises 7 (Page 49)
Exercises 8 (Page 49)
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