GMAT Math : Solving equations

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Example Question #21 : Solving Equations

Give the solution set of the equation:

$2x + \left | x\right | = 60$

Possible Answers:

x = -60

$x= -60 \textrm{ or } x = 20$

x = 60

$x = 20 \textrm{ or }x= 60$

x = 20

Correct answer:

x = 20

Explanation:

Case 1: $x \geq 0$

Then $\left | x\right | = x$ , and the equation becomes

2x + x= 60

3x= 60

$3x \div 3 = 60 \div 3$

x = 20

Case 2: x < 0

Then $\left | x\right | = -x$ , and the equation becomes

$2x +\left ( - x \right ) = 60$

x = 60

However, this conflicts with the fact that x < 0 , so this is a false solution.

The only solution is therefore x = 20 .

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Example Question #21 : Equations

Give the solution set of the equation:

$x^{2} =|x| + 6$

Possible Answers:

$x \in\left \{ -2,2\right \}$

$x \in\left \{ 2,3\right \}$

$x \in\left \{ -3,-2,2,3\right \}$

$x \in\left \{ -3,-2\right \}$

$x \in \left \{ -3,3\right \}$

Correct answer:

$x \in \left \{ -3,3\right \}$

Explanation:

Case 1: is nonnegative - that is, |x| = x .

The equation becomes

$x^{2} =x+ 6$

$x^{2} -x - 6=x+ 6 -x - 6$

$x^{2} -x - 6=0$

(x-3) (x+2 )=0

Either x-3 = 0 , in which case x = 3 ,

or x + 2 = 0 , in which case x = -2 , which is impossible, as we are assuming that is nonnegative.

case 2: is negative - that is, |x| = -x

The equation becomes

$x^{2} =-x+ 6$

$x^{2} + x - 6=-x+ 6 + x - 6$

$x^{2} + x - 6=0$

(x-2) (x+3 )=0

Either x-2= 0 , in which case x = 2 , which is impossible since we are assuming that is negative,

or x + 3 = 0 , in which case x = -3 .

x = -3 and x = 3 can both be confirmed as solutions.

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Example Question #23 : Solving Equations

What is the equation of the line that goes through (1,2) and is parallel to y=3x+4 ?

Possible Answers:

y=3x+3

y=3x+1

y=3x-4

y=3x-1

Correct answer:

y=3x-1

Explanation:

First, we have to find the slope from y=3x+4 using the general form y=mx+b. Therefore m=3

Parallel lines have the same slope, so the new line has a slope of and point (1,2) .

Use the point-slope equation: $y-y_{1}=m(x-x_{1})$

y-2=3(x-1)

y-2=3x-3

y=3x-1

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Example Question #24 : Solving Equations

What is the -intercept of 5y+2x=4 ?

Possible Answers:

(0,4)

(4,0)

(0,2)

(2,0)

Correct answer:

(2,0)

Explanation:

To solve for the -intercept, you set the to zero and solve for :

5y+2x=4

5(0)+2x=4

2x=4

x=2

-intercept: (2,0)

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Example Question #1421 : Problem Solving Questions

What is the y-intercept of 5y+2x=4 ?

Possible Answers:

(5,0)

$\left(\frac{4}{5},0\right)$

$\left(0,\frac{4}{5}\right)$

(0,5)

Correct answer:

$\left(0,\frac{4}{5}\right)$

Explanation:

To solve for the y-intercept, you set to zero and solve for :

5y+2x=4

5y+2(0)=4

5y=4

$y=\frac{4}{5}$

Therefore, the y-intercept is: $\left(0,\frac{4}{5}\right)$

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Example Question #1422 : Problem Solving Questions

What is the -intercept of y=3x-1 ?

Possible Answers:

(0,3)

$\left(0,\frac{1}{3}\right)$

$\left(\frac{1}{3},0\right)$

(3,0)

Correct answer:

$\left(\frac{1}{3},0\right)$

Explanation:

To solve for the -inter, you set to zero and solve for :

y=3x-1

0=3x-1

1=3x

$x=\frac{1}{3}$

x-intercept:

$\left(\frac{1}{3},0\right)$

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Example Question #1423 : Problem Solving Questions

What is the y-intercept of y=3x-1 ?

Possible Answers:

(1,0)

(0,1)

(0,-1)

(-1,0)

Correct answer:

(0,-1)

Explanation:

To solve for the y-intercept of y=3x-1 , you set to zero and solve for :

y=3x-1

y=3(0)-1

y=-1

y-intercept: (0,-1)

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Example Question #1424 : Problem Solving Questions

To convert Fahrenheit temperature to the equivalent in Celsius , use the formula

$C = \frac{5}{9}(F-32)$

To the nearest tenth of a degree, convert $70 ^{\circ } F$ to degrees Celsius.

Possible Answers:

$70.9^{\circ }C$

$6.9^{\circ }C$

$87.8 ^{\circ }C$

$21.1 ^{\circ }C$

$158.0 ^{\circ }C$

Correct answer:

$21.1 ^{\circ }C$

Explanation:

$C = \frac{5}{9}(F-32) = C = \frac{5}{9}(70-32) = \frac{5}{9}\cdot 38 \approx 21.1 ^{\circ }C$

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Example Question #1425 : Problem Solving Questions

Solve for :

| 0.25 - 0.4 x | = 0.1

Possible Answers:

$x = -0.875\textrm{ or } x = 0.875$

$x =- 0.375\textrm{ or }x = 0.375$

$x = -0.875\textrm{ or } x = -0.375$

$x = 0.375\textrm{ or }x = 0.875$

The equation has no solution.

Correct answer:

$x = 0.375\textrm{ or }x = 0.875$

Explanation:

Rewrite as a compound equation, then solve each equation individually:

| 0.25 - 0.4 x | = 0.1

$0.25 - 0.4 x= - 0.1 \textrm{ or }0.25 - 0.4 x= 0.1$

0.25 - 0.4 x= - 0.1

0.25 - 0.4 x- 0.25 = - 0.1 - 0.25

- 0.4 x = - 0.35

$- 0.4 x \div \left ( - 0.4 \right ) = - 0.35 \div \left ( - 0.4 \right )$

x = 0.875

0.25 - 0.4 x= 0.1

0.25 - 0.4 x- 0.25 = 0.1 - 0.25

- 0.4 x = - 0.15

$- 0.4 x \div \left ( - 0.4 \right ) = - 0.15 \div \left ( - 0.4 \right )$

x = 0.375

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Example Question #1426 : Problem Solving Questions

Solve for :

$\frac{8 }{x^{2}} - \frac{13}{x} = 6$

Possible Answers:

$x= -\frac{3}{8 }\textrm{ or }x = 2$

The equation has no solution.

$x = - \frac{1}{2} \textrm{ or } x= \frac{8 }{3}$

$x = -2 \textrm{ or } x= \frac{3}{8 }$

$x= -\frac{8 }{3}\textrm{ or }x = \frac{1}{2}$

Correct answer:

$x= -\frac{8 }{3}\textrm{ or }x = \frac{1}{2}$

Explanation:

Eliminate the denominators by multiplying by $x^{2}$ , then solve the resulting equation:

$\frac{8 }{x^{2}} - \frac{13}{x} = 6$

$\left (\frac{8 }{x^{2}} - \frac{13}{x} \right )\cdot x ^{2} = 6 \cdot x ^{2}$

$\frac{8 }{x^{2}} \cdot\frac{ x ^{2}}{1} - \frac{13}{x} \cdot\frac{ x ^{2}}{1} = 6 x ^{2}$

$8 - 13x= 6 x ^{2}$

$8 - 13x+ 13 x - 8 = 6 x ^{2} + 13 x - 8$

$6 x ^{2} + 13 x - 8 = 0$

Solve using the method:

$6 x ^{2} -3x + 16 x - 8 = 0$

$3x \left ( 2 x - 1 \right ) + 8 \left ( 2 x - 1 \right )= 0$

$\left ( 3x+ 8 \right ) \left ( 2 x - 1 \right )= 0$

$3x+ 8 = 0 \textrm{ or }2x-1 = 0$

3x+8 = 0

3x+8 -8 = 0 -8

3x = -8

$3x\div 3 = -8 \div 3$

$x= -\frac{8 }{3}$

2x-1 = 0

2x-1+1 = 0+1

2x = 1

$2x \div 2 = 1 \div 2$

$x = \frac{1}{2}$

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GMAT Math : Solving equations

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #21 : Solving Equations

Example Question #21 : Equations

Example Question #23 : Solving Equations

Example Question #24 : Solving Equations

Example Question #1421 : Problem Solving Questions

Example Question #1422 : Problem Solving Questions

Example Question #1423 : Problem Solving Questions

Example Question #1424 : Problem Solving Questions

Example Question #1425 : Problem Solving Questions

Example Question #1426 : Problem Solving Questions

All GMAT Math Resources

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