PA.8C↗️, SEC 2, inequalities with exponents (50)

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

If you’re unsure, start by temporarily treating any inequality (a comparison using <, >, ≤, or ≥) as an equation.

Solve that equation to find the boundary points where the expression equals zero.

If possible, factorise (rewrite into two brackets), or use another valid method to find those solutions.

Draw a number line, mark the boundary points, and then decide which regions make the original inequality true by testing any convenient number in each region.

Don’t forget: if at any point you multiply or divide by a negative number, flip the direction of the inequality sign.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

When we talk about the coordinate plane (the 2D grid with x and y axes), each point is an (x, y) pair.

From PA.5, you learnt about straight-line functions like y = 2x + 3.

In PA.6, you studied quadratic functions such as y = x² – 4x + 3.

A function shown as y = something draws a single line (for linear) or a curve (for quadratic) where each point on that boundary satisfies the equation (meaning its x and y values perfectly fit the formula). Here, “boundary” refers to the line or curve that divides the plane into regions where the inequality may or may not hold.

However, when you change y = … into an inequality (y >, y <, y ≥, or y ≤), it transforms from just that boundary into a shaded region (the set of all points making the inequality true).

If the boundary is dashed (┉), it’s a strict inequality (y > or y <), so those points on the line or curve aren’t included.

If the boundary is solid (━), it’s non-strict (y ≥ or y ≤), so points on the line or curve are included.

Because you’re shading an area, there can be infinitely many (x, y) possibilities for each x, as long as they fit the inequality’s rule (above, below, or on the boundary depending on the sign).

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

👀 When you see an inequality involving a squared variable (like x² ≤ some number), start by replacing the inequality sign with = to find boundary values.

Take the square root of both sides to find the crucial points, remembering that squaring a negative still gives a positive.

Next, pop the original inequality back in (≤, ≥, etc.) and decide which region (between or outside the boundaries) satisfies it.

If you multiply or divide by a negative number during any steps, flip (reverse) the inequality direction!

You’ve got this! 🤗🌟 Just work through each move carefully and watch how the inequality symbols behave.

Hint

If you see something like a² – 9 < 0, add 9 to both sides to get a² < 9, which means the solution lies between -3 and 3.

For expressions like c² – c – 6 > 0, you can isolate or factorise (if possible), find the boundary points by setting it to 0, then check which intervals make the inequality true.

Use a number line to test values in each region—this shows where the expression is above or below zero.

And remember: if you ever multiply or divide by a negative number, flip the direction of the inequality.