Question

1-5 lesson quizsolving equations and inequalities by graphing1. sketch a graph that you could use to solve the equation.$-2|x - 1| + 3 = \frac{2}{5}x - \frac{11}{5}$2. what is the solution to the equation in item 1?$\boldsymbol{\text{a}}$ $x = 1LXB0\boldsymbol{\text{c}}$ $x = -2, x = 3$$\boldsymbol{\text{d}}$ $x = -2, x = 1, x = 3$3. each time henry visits the art museum, he pays $15 for parking and $25 for admission. if he buys a membership for $110, parking will cost $10 and admission will be free. graph two equations that represent the situation.4. use the equations you graphed in item 3 to write an inequality that represents the number of museum visits for which the total member cost is less than the nonmember cost. what is the fewest number of visits that satisfies the inequality?inequality: _______________solution: _______________5. according to the table, which is the closest approximation of the solution to $f(x) = g(x)$?$\boldsymbol{\text{a}}$ 0.1207$\boldsymbol{\text{b}}$ 2.3435$\boldsymbol{\text{c}}$ 13.253$\boldsymbol{\text{d}}$ 13.256| $x$ | $f(x)$ | $g(x)$ || ---- | ---- | ---- || 13.252 | 0.9406 | -0.6205 || 13.253 | 0.1203 | 0.1212 || 13.254 | -0.7000 | 0.8629 || 13.255 | -1.5203 | 1.6046 || 13.256 | -2.3406 | 2.3463 || 13.257 | -3.1609 | 3.0880 |

Explanation:

Step1: Solve absolute value equation

First, rewrite the equation $-2|x - 1| + 3 = \frac{2}{5}x - \frac{11}{5}$. Multiply all terms by 5 to eliminate fractions:
$$-10|x - 1| + 15 = 2x - 11$$
Rearrange to isolate the absolute value:
$$-10|x - 1| = 2x - 26$$
$$|x - 1| = \frac{26 - 2x}{10} = \frac{13 - x}{5}$$
Split into two cases:
Case 1: $x - 1 = \frac{13 - x}{5}$
$$5(x - 1) = 13 - x$$
$$5x - 5 = 13 - x$$
$$6x = 18 \implies x = 3$$
Case 2: $x - 1 = -\frac{13 - x}{5}$
$$5(x - 1) = x - 13$$
$$5x - 5 = x - 13$$
$$4x = -8 \implies x = -2$$

Step2: Define cost equations for museum

Let $x$ = number of visits.
Non-member cost: $C_n = 15x + 25x = 40x$
Member cost: $C_m = 110 + 10x$

Step3: Set up and solve cost inequality

Find when member cost < non-member cost:
$$110 + 10x < 40x$$
$$110 < 30x$$
$$x > \frac{110}{30} \approx 3.67$$
The smallest integer $x$ is 4.

Step4: Find $f(x)=g(x)$ from table

Look for where $f(x)$ and $g(x)$ are closest in value. At $x=13.253$, $f(x)=0.1203$ and $g(x)=0.1212$, which are nearly equal.

Answer:

1. (Graph: Plot $y=-2|x-1|+3$ (V-shaped, vertex at (1,3), slope $\pm2$) and $y=\frac{2}{5}x-\frac{11}{5}$ (line with slope $\frac{2}{5}$, y-intercept $-\frac{11}{5}$); their intersections are at $x=-2$ and $x=3$)
2. C. $x=-2, x=3$
3. (Graph: $y=40x$ (line through origin, slope 40) and $y=110+10x$ (line with y-intercept 110, slope 10))
4. inequality: $110 + 10x < 40x$

solution: 4

1. C. 13.253