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Grade 9 · Inequalities

Motion tracker: Test a complete solution ray

Motion tracker investigation: test a complete solution ray. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Motion tracker: Test a complete solution ray. Boundary x: -1. Solution: x ≥ -1. Moving test input: -12. Test satisfies inequality: No-1x ≤ 1 · shaded ray includes equality-30-1501530x ≥ -1Amber probe: satisfies · navy: outside
The sign control accepts −1 or +1 and the magnitude stays positive, so division by zero is excluded. The number line window is −30 to 30; rays continue beyond it. Equality is always included.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Boundary x
-1
Solution
x ≥ -1
Moving test input
-12
Test satisfies inequality
No

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

An inequality describes a set of inputs rather than just one answer. Dividing by a negative coefficient reverses the direction of comparison because it reverses order on the number line. The boundary remains included for a non-strict inequality. This investigation begins with coefficient magnitude = 1; right-side bound = 1; coefficient sign: −1 or +1 = -1. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

ax ≤ b; x ≤ b/a if a>0, x ≥ b/a if a<0

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Form a by multiplying its positive magnitude by the selected sign, then locate the boundary b/a.

  2. STEP 2

    Follow the changing model

    A filled boundary dot includes equality; the shaded ray shows the permitted side and continues beyond the visible window.

  3. STEP 3

    Check and explain the relationship

    Move the test input through the line and substitute it into ax≤b. Its membership should agree with the shaded ray.

Your turn to explain

Make a prediction. Test your reasoning.

Motion tracker: Solve -1x ≤ 1, including the boundary.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

x ≥ -1. Dividing by a negative coefficient reverses the sign.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 9 · Algebra

Clear fractions in a linear inequality

Multiply by a positive common denominator before solving a linear inequality.