Algebra II - Simplifying Inequalities

Consider the inequality

$\left | y+9 \right | > 5$

Which of the following statements has the same solution set?

$y < -14 \textup{ or } y > -4$

$y < -14 \textup{ or } y > 4$

$y < -14 \textup{ or } y > 14$

$y < - 4 \textup{ or } y > 4$

$y < - 4 \textup{ or } y > 1 4$

Explanation

$\left | y+9 \right | > 5$

can be written as the compound inequality statement

$y+9 <-5 \textup{ or } y + 9 > 5$

In each expression, 9 can be subtracted to yield the inequality statement

$y < -14 \textup{ or } y > -4$

This is the correct response.

Simplify the following inequality:

$-4(2x-2)\geq x+14$

$x\leq -\frac{2}{3}$

$x\geq -\frac{2}{3}$

$x\geq -\frac{6}{7}$

$x\leq -\frac{6}{7}$

$x\leq 1$

Explanation

First distribute the -4 to give:

$-8x+8\geq x+14$

Subtract x from both sides to get x on one side to give:

$-9x+8\geq 14$

Subtract 8 from both sides to get x term by itself to give:

$-9x\geq 6$

Divide both sides by -9, and remember when dividing or multiplying both sides by a negative it changes the inequality to give:

$x\leq -\frac{6}{9}$

Simplify fraction to give final answer:

$x\leq -\frac{2}{3}$

Consider the following inequality:

$\left | x-10\right |>5$

Which of the following statements has the same solution set?

and

Explanation

$\left | x-10\right |>5$

1. Rewrite as a compound inequality statement:

2. Simplify:

becomes

Consider the inequality

$\left | x - 7 \right | < 16$

Which of the following statements has the same solution set?

Explanation

$\left | x - 7 \right | < 16$

is equivalent to the three-way inequality

Add 7 to all three expressions to yield the statement:

This is the correct response.

Consider the following equality:

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

Which of the following gives the same solution set?

Explanation

$\left |x-7 \right |>4$ Can be written as a compound inequality statement:

Solving these inequalities gives:

Write and simplify the inequality: Two times the quantity of two less than a number squared is less than four.

Explanation

Write the inequality in parts before simplifying.

A number squared:

Two less than a number squared:

Two times the quantity of two less than a number squared:

Is less than four:

Divide by two on both sides.

$\frac{2(x^2-2)}{2}<\frac{4}{2}$

Simplify both sides.

Add two on both sides.

Simplify both sides.

Square root both sides.

$\sqrt{x^2}<\sqrt{4}$

$\pm x<2$

This will split into two solutions.

The first answer is:

There will be a negative component as well for the second answer.

Dividing by a negative one on both sides will result in switching the sign.

The answers are:

Which coordinate is a solution to the inequality

$y+1\leq 3x+5$

Explanation

Subtract 1 on both sides of the inequality to get

$y \leq 3x+4$

From here we plug in each set of coordinates to see which could satisfy the statement.

$\text{For } (0,2):$

$2\le3(0)+4$

$2\le4$

This is a true statement therefore we say that (0,2) satisfies the inequality.

Simplify the following inequality:

$3x-3\geq 6$

$x\geq3$

$x\geq1$

$x\geq6$

Explanation

To simplify an inequality we want to isolate the variable on one side of the inequality sign. In order to accomplish this remember to do the reverse opperation to move numbers from one side to the other.

$3x-3\geq 6$

The reverse operation of subtraction is addition. Therefore to move the three from the left hand side to the right hand side we will need to add 3 on each side of the inequality to get

$3x \geq 9$ .