Solving Equations and Inequallities

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College Algebra › Solving Equations and Inequallities

Questions 1 - 10

1

Solve:

Explanation

We need to set up two equations since we are dealing with absolute value

$\x-10=20\x-10=-20$

We solve for , in each equation to get the solutions.

2

Solve:

Explanation

We need to set up two equations since we are dealing with absolute value

$\x-10=20\x-10=-20$

We solve for , in each equation to get the solutions.

3

Express the following linear inequality in interval notation.

$\left ( -2,\infty \right )$

$\left [ -2,\infty \right ]$

$\left ( -\infty ,-2\right )$

$(-\infty ,2)$

$\left ( 2,\infty \right )$

Explanation

Upon solving for x, we find that x is larger than -2. The left-hand term of the interval is -2 since it is the lower bound for our set, and it has a parenthesis around it because it is not included in our set (-2 is not greater than -2). The right-hand term of the interval is positive infinity because any number larger than -2 is in the set. There is always a parenthesis around infinity or negative infinity.

4

Express the following linear inequality in interval notation.

$\left ( -2,\infty \right )$

$\left [ -2,\infty \right ]$

$\left ( -\infty ,-2\right )$

$(-\infty ,2)$

$\left ( 2,\infty \right )$

Explanation

Upon solving for x, we find that x is larger than -2. The left-hand term of the interval is -2 since it is the lower bound for our set, and it has a parenthesis around it because it is not included in our set (-2 is not greater than -2). The right-hand term of the interval is positive infinity because any number larger than -2 is in the set. There is always a parenthesis around infinity or negative infinity.

5

Solve the inequality:

$8+\left | 4x-7\right |\geq17$

$x\geq4:or: x\leq-\frac{1}{2}$

$x\leq 4:or: x\geq -\frac{1}{2}$

$x\geq-2$

$x\geq2:or: x\leq\frac{3}{4}$

$x\leq-\frac{1}{}5$

Explanation

To solve for first isolate the absolute value portion on one side of the equation and all other constants on the other side.

$8+\left | 4x-7\right |\geq17$

$\left | 4x-7\right |\geq9$

Since absolute values can come from either negative or positive values, two equations need to be set up and solved for.

$4x-7\geq9;or;4x-7\leq-9$

$4x\geq16;or;4x-7\leq-2$

$x\geq4:or: x\leq-\frac{1}{2}$

6

Solve the inequality:

$8+\left | 4x-7\right |\geq17$

$x\geq4:or: x\leq-\frac{1}{2}$

$x\leq 4:or: x\geq -\frac{1}{2}$

$x\geq-2$

$x\geq2:or: x\leq\frac{3}{4}$

$x\leq-\frac{1}{}5$

Explanation

To solve for first isolate the absolute value portion on one side of the equation and all other constants on the other side.

$8+\left | 4x-7\right |\geq17$

$\left | 4x-7\right |\geq9$

Since absolute values can come from either negative or positive values, two equations need to be set up and solved for.

$4x-7\geq9;or;4x-7\leq-9$

$4x\geq16;or;4x-7\leq-2$

$x\geq4:or: x\leq-\frac{1}{2}$

7

Solve:

Explanation

Move to the other side by subtracting it from both sides.

Simplify:

Divide by the coefficient, the number in front of x.

$\frac {4x}{4}=\frac {8}{4}$

Reduce:

8

Solve:

Explanation

We need to set up two equations since we are dealing with absolute value

$\x-45=2\x-45=-2$

We solve for , in each equation to get the solutions.

9

Solve:

Explanation

Move to the other side by subtracting it from both sides.

Simplify:

Divide by the coefficient, the number in front of x.

$\frac {4x}{4}=\frac {8}{4}$

Reduce:

10

Solve:

Explanation

We need to set up two equations since we are dealing with absolute value

$\x-45=2\x-45=-2$

We solve for , in each equation to get the solutions.

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