Question

1 review
name ____________ block __
when using desmos, what is the difference between solving an absolute - value equation versus solving an absolute value inequality?
what does it mean when:

• everything is shaded in on desmos when graphing an absolute value inequality?
• nothing is shaded in on desmos when graphing an absolute value inequality?

a. solve the following equations and inequalities using desmos. write what you are typing in desmos!

1. 2|4x + 1| - 5 = 1 2. 2|4b|+13 < 25 3. |2t - 3|+4 = 2 4. 5 + 8|-2n - 1| > - 7

b. write an absolute value inequality that represents the situation. |x - target|≤ or ≥ varies

1. a regulation bowling pin should weigh 56 oz. but can vary by 2 oz. what is the range of acceptable weights of a bowling pin?
2. a cookie company sells a 16 oz package of cookies that may vary by as much as an oz. what is the range of acceptable weights of cookie packages?
3. julie is 67 inches tall. she will only date guys that are within 4 inches of her height. what is a range of heights for guys that julie will not date?

c. evaluate the piece - wise functions f(x)=(\begin{cases}x^{2}+4,x < 0\sqrt{x}+4,xgeq0end{cases}) g(x)=(\begin{cases}|x|-1,x > - 1\\x + 3,xleq - 1end{cases})

1. f(16) 2. g(4) 3. f(-2)+g(-1) 4. 3g(5)-f(9)

d. fill in the chart below for each absolute value function. compare to y = |x|
1 y=-|x + 1|
vertex: (_,_)
opens: up / down
maximum/minimum of ____
narrower/wider/same
2 y = 7|x - 3|-4
vertex: (_,_)
opens: up / down
maximum/minimum of ____
narrower/wider/same
3 y=-\frac{2}{3}|x - 1|
vertex: (_,_)
opens: up / down
maximum/minimum of ____
narrower/wider/same

Explanation:

Step1: Solve absolute - value equation 2|4x + 1| - 5 = 1

Isolate the absolute - value term. First, add 5 to both sides: 2|4x + 1|=6. Then divide both sides by 2: |4x + 1| = 3. This gives two equations: 4x+1 = 3 and 4x + 1=-3. Solving 4x+1 = 3 gives 4x=2, so x = 0.5. Solving 4x + 1=-3 gives 4x=-4, so x=-1.

Step2: Solve absolute - value inequality 2|4b|+13 < 25

Isolate the absolute - value term. Subtract 13 from both sides: 2|4b|<12. Divide both sides by 2: |4b|<6. This gives - 6 < 4b < 6. Divide all parts by 4: - 1.5 < b < 1.5.

Step3: Solve absolute - value equation |2t - 3|+4 = 2

Isolate the absolute - value term: |2t - 3|=-2. Since the absolute value of a number is non - negative, there is no solution.

Step4: Solve absolute - value inequality 5 + 8|-2n - 1|>-7

Isolate the absolute - value term. Subtract 5 from both sides: 8|-2n - 1|>-12. Divide both sides by 8: |-2n - 1|>-1.5. Since the absolute value of any real number is non - negative, the solution is all real numbers, n∈R.

Step5: Write absolute - value inequality for bowling pin weight

Let x be the weight of the bowling pin. The target weight is 56 oz and the variation is 2 oz. The absolute - value inequality is |x - 56|≤2. Solving this gives 54≤x≤58.

Step6: Write absolute - value inequality for cookie package weight

Let x be the weight of the cookie package. The target weight is 16 oz and the variation is 1 oz. The absolute - value inequality is |x - 16|≤1. Solving this gives 15≤x≤17.

Step7: Write absolute - value inequality for Julie's dating situation

Let x be the height of the guy. The target height relative to Julie is 67 inches and the acceptable difference is 4 inches. The inequality for acceptable heights is |x - 67|≤4, and the inequality for non - acceptable heights is |x - 67|>4.

Step8: Evaluate piece - wise function f(16)

Since x = 16≥0, for f(x)=\begin{cases}x^{2}+4,x < 0\\\sqrt{x}+4,x\geq0\end{cases}, we use f(x)=\sqrt{x}+4. So f(16)=\sqrt{16}+4=4 + 4=8.

Step9: Evaluate piece - wise function g(4)

Since x = 4>-1, for g(x)=\begin{cases}|x|-1,x > - 1\\x + 3,x\leq-1\end{cases}, we use g(x)=|x|-1. So g(4)=|4|-1=4 - 1=3.

Step10: Evaluate f(-2)+g(-1)

Since x=-2 < 0, for f(x), f(-2)=(-2)^{2}+4=4 + 4=8. Since x=-1, for g(x), g(-1)=-1 + 3=2. So f(-2)+g(-1)=8 + 2=10.

Step11: Evaluate 3g(5)-f(9)

Since x = 5>-1, for g(x), g(5)=|5|-1=5 - 1=4. So 3g(5)=3×4 = 12. Since x = 9≥0, for f(x), f(9)=\sqrt{9}+4=3 + 4=7. So 3g(5)-f(9)=12-7 = 5.

Step12: Analyze absolute - value function y=-|x + 1|

The vertex of y=-|x + 1| is (-1,0). Since the coefficient of |x| is negative, it opens down. The maximum value is 0. Compared to y = |x|, it is the same width.

Step13: Analyze absolute - value function y = 7|x - 3|-4

The vertex of y = 7|x - 3|-4 is (3,-4). Since the coefficient of |x| is positive, it opens up. The minimum value is - 4. Compared to y = |x|, it is narrower because the coefficient 7>1.

Step14: Analyze absolute - value function y=-\frac{2}{3}|x - 1|

The vertex of y=-\frac{2}{3}|x - 1| is (1,0). Since the coefficient of |x| is negative, it opens down. The maximum value is 0. Compared to y = |x|, it is wider because 0<\frac{2}{3}<1.

Difference between solving absolute - value equation and inequality

When solving an absolute - value equation like |ax + b|=c (c≥0), we set up two equations ax + b=c and ax + b=-c. When solving an absolute - value inequality like |ax + b|<c (c>0), we set up - c<ax + b<c, and for |ax + b|>c (c>0), we set up ax + b>c or ax + b<-c.

Meaning of shading in Desmos for absolute - value inequality

If everything is shaded in when graphing an absolute - value inequality in Desmos, it means the solution set is all real numbers (e.g., for |ax + b|>-k where k>0). If nothing is shaded in, it means there is no solution (e.g., for |ax + b|<-k where k>0).

Answer:

1. x = 0.5 or x=-1
2. - 1.5 < b < 1.5
3. No solution
4. n∈R
5. |x - 56|≤2, 54≤x≤58
6. |x - 16|≤1, 15≤x≤17
7. |x - 67|>4
8. f(16)=8
9. g(4)=3
10. f(-2)+g(-1)=10
11. 3g(5)-f(9)=5
12. Vertex: (-1,0), Opens: DOWN, MAXIMUM of 0, SAME width as y = |x|
13. Vertex: (3,-4), Opens: UP, MINIMUM of - 4, NARROWER than y = |x|
14. Vertex: (1,0), Opens: DOWN, MAXIMUM of 0, WIDER than y = |x|

Difference: Equation: set up two equations; Inequality: set up compound inequalities
Everything shaded: solution is all real numbers; Nothing shaded: no solution