section 4: evaluating & simplifying expressions (calculators not allowed)
24. add: 7/12 + 5/18 = ______ (simplest form)

25. divide: 5/6 ÷ 3/4 = ______

26. solve: (2/5)x = 20 x = ______

27. simplify completely: √200 =

28. multiply: (3√5)(4√10) = ______
29. divide: (2√3)/√6 = ______
section 5: solving equations & graphing linear functions (calculators not allowed)
30. solve: 3(x − 4) = 2x + 7

31. solve: |x − 5| = 9

32. solve: −4(x + 3) + 2x = −18

33. write the equation of a vertical line through (−2, 7)

34. write the equation of a horizontal line through (0, −6)

35. find x- and y-intercepts of 3x − 4y = 24

36. graph 2x + 3y = 12 using intercepts method (label both intercepts)

37. write the equation in slope-intercept form: passes through (0, −5), slope = 3

Question

section 4: evaluating & simplifying expressions (calculators not allowed)
24. add: 7/12 + 5/18 = ______ (simplest form)

25. divide: 5/6 ÷ 3/4 = ______

26. solve: (2/5)x = 20  x = ______

27. simplify completely: √200 =

28. multiply: (3√5)(4√10) = ______
29. divide: (2√3)/√6 = ______
    section 5: solving equations & graphing linear functions (calculators not allowed)
30. solve: 3(x − 4) = 2x + 7

31. solve: |x − 5| = 9

32. solve: −4(x + 3) + 2x = −18

33. write the equation of a vertical line through (−2, 7)

34. write the equation of a horizontal line through (0, −6)

35. find x- and y-intercepts of 3x − 4y = 24

36. graph 2x + 3y = 12 using intercepts method (label both intercepts)

37. write the equation in slope-intercept form: passes through (0, −5), slope = 3

Accepted Answer

### Question 24:
# Explanation:
## Step1: Find common denominator
The denominators are 12 and 18. The least common multiple of 12 and 18 is 36.
## Step2: Rewrite fractions
Rewrite $\frac{7}{12}$ as $\frac{7\times3}{12\times3}=\frac{21}{36}$ and $\frac{5}{18}$ as $\frac{5\times2}{18\times2}=\frac{10}{36}$.
## Step3: Add the fractions
$\frac{21}{36}+\frac{10}{36}=\frac{21 + 10}{36}=\frac{31}{36}$
# Answer:
$\frac{31}{36}$

### Question 25:
# Explanation:
## Step1: Recall division rule for fractions
Dividing by a fraction is multiplying by its reciprocal. So $\frac{5}{6}\div\frac{3}{4}=\frac{5}{6}\times\frac{4}{3}$.
## Step2: Multiply numerators and denominators
Multiply numerators: $5\times4 = 20$, multiply denominators: $6\times3=18$. So we get $\frac{20}{18}$.
## Step3: Simplify the fraction
Simplify $\frac{20}{18}$ by dividing numerator and denominator by 2: $\frac{10}{9}$ or $1\frac{1}{9}$
# Answer:
$\frac{10}{9}$ (or $1\frac{1}{9}$)

### Question 26:
# Explanation:
## Step1: Isolate $x$
To solve $\frac{2}{5}x=20$, multiply both sides by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$.
## Step2: Calculate $x$
$x = 20\times\frac{5}{2}$. $20\div2 = 10$, $10\times5=50$. So $x = 50$.
# Answer:
$50$

### Question 27:
# Explanation:
## Step1: Factor the number inside the square root
Factor 200 as $100\times2$. So $\sqrt{200}=\sqrt{100\times2}$.
## Step2: Use square - root property
Using the property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ ($a\geq0,b\geq0$), we have $\sqrt{100\times2}=\sqrt{100}\times\sqrt{2}$.
## Step3: Simplify $\sqrt{100}$
Since $\sqrt{100} = 10$, then $\sqrt{200}=10\sqrt{2}$ (Note: The original answer of $2\sqrt{10}$ is incorrect. The correct simplification is $10\sqrt{2}$)
# Answer:
$10\sqrt{2}$

### Question 28:
# Explanation:
## Step1: Multiply the coefficients and the square - roots separately
For $(3\sqrt{5})(4\sqrt{10})$, multiply the coefficients $3\times4 = 12$ and multiply the square - roots $\sqrt{5}\times\sqrt{10}=\sqrt{5\times10}=\sqrt{50}$.
## Step2: Simplify $\sqrt{50}$
Factor 50 as $25\times2$, so $\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$.
## Step3: Multiply the result from step 1 and step 2
$12\times5\sqrt{2}=60\sqrt{2}$
# Answer:
$60\sqrt{2}$

### Question 29:
# Explanation:
## Step1: Rationalize the denominator
To divide $\frac{2\sqrt{3}}{\sqrt{6}}$, multiply numerator and denominator by $\sqrt{6}$. So we get $\frac{2\sqrt{3}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}$.
## Step2: Simplify the numerator and denominator
Simplify the numerator: $\sqrt{3}\times\sqrt{6}=\sqrt{3\times6}=\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}$, so the numerator becomes $2\times3\sqrt{2}=6\sqrt{2}$. The denominator is $\sqrt{6}\times\sqrt{6}=6$.
## Step3: Simplify the fraction
$\frac{6\sqrt{2}}{6}=\sqrt{2}$
# Answer:
$\sqrt{2}$

### Question 30:
# Explanation:
## Step1: Expand the left - hand side
Expand $3(x - 4)$ using the distributive property: $3x-12$. So the equation becomes $3x - 12=2x + 7$.
## Step2: Isolate $x$
Subtract $2x$ from both sides: $3x-2x-12=2x-2x + 7$, which simplifies to $x-12 = 7$. Then add 12 to both sides: $x-12 + 12=7 + 12$.
## Step3: Solve for $x$
We get $x = 19$
# Answer:
$x = 19$

### Question 31:
# Explanation:
## Step1: Use the absolute - value property
If $|x - 5|=9$, then $x - 5=9$ or $x - 5=-9$.
## Step2: Solve the two equations
For $x - 5=9$, add 5 to both sides: $x=9 + 5=14$. For $x - 5=-9$, add 5 to both sides: $x=-9 + 5=-4$.
# Answer:
$x = 14$ or $x=-4$

### Question 32:
# Explanation:
## Step1: Expand the left - hand side
Expand $-4(x + 3)$ using the distributive property: $-4x-12$. So the equation becomes $-4x-12 + 2x=-18$.
## Step2: Combine like terms
Combine $-4x$ and $2x$: $-2x-12=-18$.
## Step3: Isolate $x$
Add 12 to both sides: $-2x-12 + 12=-18 + 12$, which gives $-2x=-6$. Then divide both sides by $-2$: $x = 3$
# Answer:
$x = 3$

### Question 33:
# Explanation:
## Step1: Recall the equation of a vertical line
A vertical line has an undefined slope and its equation is of the form $x = a$, where $a$ is the $x$ - coordinate of any point on the line.
## Step2: Determine the equation
The line passes through $(-2,7)$, so the $x$ - coordinate is $-2$. Thus, the equation of the vertical line is $x=-2$
# Answer:
$x=-2$

### Question 34:
# Explanation:
## Step1: Recall the equation of a horizontal line
A horizontal line has a slope of 0 and its equation is of the form $y = b$, where $b$ is the $y$ - coordinate of any point on the line.
## Step2: Determine the equation
The line passes through $(0,-6)$, so the $y$ - coordinate is $-6$. Thus, the equation of the horizontal line is $y=-6$
# Answer:
$y=-6$

### Question 35:
# Explanation:
## Step1: Find the $x$ - intercept
To find the $x$ - intercept, set $y = 0$ in the equation $3x-4y = 24$. Then $3x-4(0)=24$, which simplifies to $3x=24$. Divide both sides by 3: $x = 8$. So the $x$ - intercept is $(8,0)$.
## Step2: Find the $y$ - intercept
To find the $y$ - intercept, set $x = 0$ in the equation $3x-4y = 24$. Then $3(0)-4y=24$, which simplifies to $-4y=24$. Divide both sides by $-4$: $y=-6$. So the $y$ - intercept is $(0,-6)$
# Answer:
$x$ - intercept: $(8,0)$, $y$ - intercept: $(0,-6)$

### Question 36:
# Explanation:
## Step1: Find the $x$ - intercept
Set $y = 0$ in $2x + 3y=12$. Then $2x+3(0)=12$, so $2x = 12$ and $x = 6$. The $x$ - intercept is $(6,0)$.
## Step2: Find the $y$ - intercept
Set $x = 0$ in $2x + 3y=12$. Then $2(0)+3y=12$, so $3y = 12$ and $y = 4$. The $y$ - intercept is $(0,4)$.
## Step3: Graph the line
Plot the points $(6,0)$ and $(0,4)$ on the coordinate plane and draw a straight line through them. Label the $x$ - intercept as $(6,0)$ and the $y$ - intercept as $(0,4)$
# Answer:
The line passes through $(6,0)$ (x - intercept) and $(0,4)$ (y - intercept), and a straight line is drawn through these two points.

### Question 37:
# Explanation:
## Step1: Recall the slope - intercept form
The slope - intercept form of a line is $y=mx + b$, where $m$ is the slope and $b$ is the $y$ - intercept.
## Step2: Determine $m$ and $b$
We know that the slope $m = 3$ and the line passes through $(0,-5)$. The $y$ - intercept $b$ is the $y$ - coordinate of the point where $x = 0$, so $b=-5$.
## Step3: Write the equation
Substitute $m = 3$ and $b=-5$ into $y=mx + b$. We get $y = 3x-5$
# Answer:
$y = 3x-5$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 24:

Step1: Find common denominator

Step2: Rewrite fractions

Step3: Add the fractions

Step1: Recall division rule for fractions

Step2: Multiply numerators and denominators

Step3: Simplify the fraction

Step1: Isolate \(x\)

Step2: Calculate \(x\)

Answer:

Question 25: